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AUTHOR Yates, Robert C. TITLE Curves and Their Properties. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 74 NOTE 259p.; Classics in Mathematics Education, Volume 4 AVAILABLE FROM National Council of Teachers of Mathematics, Inc., 1906 Association Drive, Reston, Virginia 22091 ($6.40)

EDRS PRICE MF-$0.75 HC Not Available from EDRS. PLUS POSTAGE DESCRIPTORS Analytic Geometry; *College Mathematics; Geometric Concepts; *Geometry; *Graphs; Instruction; Mathematical Enrichment; Mathematics Education;Plane Geometry; *Secondary School Mathematics IDENTIFIERS *Curves

ABSTRACT This volume, a reprinting of a classic first published in 1952, presents detailed discussions of 26 curves or families of curves, and 17 analytic systems of curves. For each curve the author provides a historical note, a sketch orsketches, a description of the curve, a a icussion of pertinent facts,and a bibliography. Depending upon the curve, the discussion may cover defining equations, relationships with other curves(identities, derivatives, integrals), series representations, metricalproperties, properties of tangents and normals, applicationsof the curve in physical or statistical sciences, and other relevantinformation. The curves described range from thefamiliar conic sections and trigonometric functions through tit's less well knownDeltoid, Kieroid and Witch of Agnesi. Curve related - systemsdescribed include envelopes, evolutes and pedal curves. A section on curvesketching in the coordinate plane is included. (SD) U S DEPARTMENT OFHEALTH. EDUCATION II WELFARE NATIONAL INSTITUTE OF EDUCATION THIS DOCuME N1 ITASOLE.* REPRO MAE° EXACTLY ASRECEIVED F ROM THE PERSON ORORGANI/AlICIN ORIGIN ATING 11 POINTS OF VIEWOH OPINIONS STATED DO NOT NECESSARILYREPRE INSTITUTE OF SENT OFFICIAL NATIONAL EDUCATION POSITION ORPOLICY

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1 l' 41 CURVES AND THEIR PROPERTIES CLASSICS IN MATHEMATICS EDUCATION

Volume 1: The Pythagorean haposition by Elisha Scott Loomis Volume 2: Number Stories of Long Ago by David Eugene Smith Volume 3: The Trisection Problem by Robert C. Yates Volume 4: Curves and Their Properties by Robert C. Yates RIIRIABLE BESICOP1

CONES AID THE II PROPERTIES

OW CAMS

Library of Congress Catalog Card Number: 7.140222 Printed in the United States of America 1974

tV FOREWORD

Some mathematical works of considerable vintage havea timeless quality about them. Like classics inany field, they still bring joy and guidance to the reader. Books of this kind, ifno longer readily available, are being sought out by the National Council of Teachers of Mathematics, which has begun to publisha series of such classics. The present title is the fourth volume of the series. A Handbook on Curves and Their Propertieswas first published in 1952 when the author was teaching at the United States Military Academyat West Point. A photoithoprint reproductionwas issued in 1959 by Edwards Brothers, Inc., Lithoprinters, of Ann Arbor,Michigan. The present reprint edition has been similarly produced, by photo-offset, from a copy of the 1959 edition. Except for providing new front matter, including a biographical sketch of the author and this Foreword byway of, explanation, no attempt has been made to modernize the bookin any way. To do so would surely detract from, rather than add to, its value. ABOUT THE AUTHOR

Robert Carl Yates was born in .Falls Church, Virginia,on 10 March 1904. In 1924 he receiveda B.S. degree in civil engineering from Virginia Military Institute. This degreewas followed by an A.B. degree in psy- chology and education from Washington and LeeUniversity in 1926 and by the M.A. and Ph.D. degrees in mathematics andapplied mathematics from Johns Hopkins University in 1928 and 1930. While working on these later degrees Bob Yateswas an instructor at Virginia Military Institute, the University of Maryland,and Johns Hop- kins University. After completing his Ph.D.degree, he accepted a Posi- tion as assistant professor, in 1931, at the University of Maryland,where later he was promoted to associate professor. In 1939 hebecame associate professor of mathematics at Louisiana StateUniversity. As a captain in the Army Reserves, Professor Yates reportedto the United States Military Academy for active dutyon 6 June 1942. Before leaving the Academy herose to the rank of colonel and the title of associate professor of mathematics. He left West Point in August 1954, whena reduction in the number of colonels was authorized at USMA, and accepteda position as professor of mathematics at Virginia PolytechnicInstitute.In 1955 he became professor of mathematics and chairman of the departmentat the College of William and Mary. The last position he heldwas as one of the original professors at the University of South Florida, beginningin 1960. He went to this rew institution as chairman of the Department of Mathematics, resigning the chairmanship in 1962 in order to devotemore time to teaching, lecturing, and writing. During his tour of duty at West Point, Dr. Yatesspent many of his summers as a visiting professor. Among the institutions he served were Teachers College, Columbia University; YeshivaUniversity; and johns Hopkins University. Robert Yates wasa man of many talents. Although he was trained in pure and applied mathematics, he became interested In the field of mathematics education rather early in his professionqlcareer. In both vii viii AuouT THE AUTHOR al'eaS he Wilt lip a (Ilse reputation as alecturer and a writer. During his lifetime he had sixty-odd papers published in various researchand mathematical-education journals, including the Mathematics Teacher, and in NCTM yearbooks. He also wrote five books dealing with various aspects of geometry, the calculus. and differential equations. From1937 until he was called to active duty atWest Point in 1942, he served on the editorial hoard, and as editor of one department, of the NationalMathematics Magazine. These were some of his professional achievements. Ms activities,how ever, were not limited to theworld of mathematics.At VM1where he was a member of thetrack squad, dramatics and journalism claimed some of his time. Music became a continuing resource. In laterlife his recrea- tions included playing the piano as well as sailing,skating, and golf. Dr. Yates, whose social fraternity was Kappa Alpha, waselected to two scientific honor societies: Gamma Alpha and Sigma Xi. Holding member- ship in the American Mathematical Society, the Mathematical Association of America, and the National Council of Teachersof Mathematics, he was at one period a governor ofthe MAA. He was also a member of the MAA's original ad hoc Committee on the UndergraduateProgram in Mathematics (CUPM ).In late 1961 he was selected by the Association of Higher Education as one of twenty-five"outstanding college andUni- versity educators in America today," and on 4February 1962 he was featured on the ABC-TV program "Meet the Professor." Dr. Yates had been interested in mathematicseducation before 1939. Howcwer, when he came to Louisiana State University, his work inthis field began to expand. Owing to his efforts, the Departmentof Mathe- matics and the College of Education made some importantchanges in the mathematical curriculum for the training of prospective secondaryschool teachers. One of the most important additions was six semesterhours in geometry. Dr. Yates was given this course toteach, and for a text he used his first book, Geometrical Tools, From this beginninghis interest and work in mathematics education increased, while hecontinued to lecture and write in the areas of pure and applied mathematics, The atmosphere at Nk'est Point was quite a change for Dr.Yates. How- ever, even here he continuedhis activities in mathematics education. One of his duties was to supervise and conduct courses inthe techniques of teaching mathematics. These were courses designed forthe groups of new instructors whojoined the departntme staff annually; for most of the faculty at the Academy, then as now, were active-duty officerswho came on a first or second tour ofthree to four years' duration. In performing this duty he was considered a superior instructor and also anexcellent teacher of teachers. ABOUT THE AUTHOR ix

After leaving the service Professor Yates continued his effortsto improve mathematics education. During thesummers he taught in several different mathematics institutes, and hewas a guest lecturer in many summer and academic-year institutes supported by the National Science Foundation. In earlier years he both taught and lectured in the grandfather of all institutes, the one developed by Professor W. W. Rankin at Duke Uni- versity.In Virginia and later in Florida he servedas a consultant to teachers of mathematics in various school districts. During the academic years 1961/62 and 1962/63 the University of South Florida was engaged in an experimental television program. Professor Yateswas the television lecturer in the course materials deNeloped through thisprogram. As a result of this programas well as the MAA lectureship program for high schools, supported by he NSF, he traveledto all parts of Florida giving lectures and consulting with high school teachers. Through all these activities Dr. Yates greatly enhancedthe field of mathematics education. He builtup a reputation as an outstanding lecturer with a pleasing, interest-provoking presentation anda rare ability to talk while illustrating his subject. Those who have heard him will long remember bhp and his great ability. Others will find thathis writ- ings show, soinevAat vicariously, thesesame characteristics. By his first wife, Naomi Sherman, who died in childbirth, he had three children. Robert Jefferson, the eldest, isnow in business in California. Melinda Susan, the youngest, isnow Mrs. Richard B. Shaw, the wife of a Missouri surgeon. Mrs. Shaw majored in mathematics at Mount Holyoke College and works as a computerprogrammer and systems analyst. The second child is Daniel Sherman. He is following in his father's footsteps and has completed his doctorate in mathematics education at Florida State University. He is a mathematics specialist at the Mathematics and Science Center, a resource center serving the public schools in the city of Richmond, Virginia, and in the counties of Chesterfield, Cooehland, Henrico, and Powhatan. Dr. Yates passed away on 18 December 1963 andwas interred in Arlington National Cemetery. HOUSTON T. KARNES Louisiana State University NOTATION x,y = Rectangular Coordinates.

9,r Polar Coordinate, (Radius Vector).

U. Parameter or Polar Coordinate.

4 Inclination of Tangent.

41. Angle between a Tangent and the Radiun Vectorto Point of Tangency.

Arc Length.

. Are of Evoluto (or Standard Deviation).

p Distance from Origin to Tangent.

. Length; A Area; V Volume; X = Surface Area.

. Surface of Revolution about theX-ratio.

Vx . Volume of Revolution about the X-axis.

N . Normal Length.

R = Radius of Curvature.

K = Curvature.

v . Velocity; a = Acceleration. a . = (t . Time or a Parameter). dx dt

dF , x , t(or -&-)

z = x + iy, a Complex Variable.

f(s4) = 0: The Whewell intrinsic Equation.

f(170) . 0: The Cestiro Intrinsic Equation.

f(r,p) = 0: The Pedal Equation. CONTENTS

Page

Astroid 1

Cardioid 4 Cassinian Curves 8 Catenary 12 Caustics 15 Circle 21 Cissoid 26 Conchoid 31 Cones 34 Conics 36 Cubic Parabola 56 Curvature 60 Cycloid 65 Deltoid 71

Envelopes 75 Epi- and Hypo-Cycloids 81 Evolutes 86 Exponential Curves 93 Polium of Descartes 98 Functions with Discontinuous Properties 100 Olissettes 108 Hyperbolic Functions 113 Instantaneous Center of Rotation and the Construction of Some Tangents 119 Intrinsic Equations 123 Inversion 127 Involutes 135 Isoptic Curves 138 Kieroid 141 Lemniscate of Bernoulli 143 Limacon of Pascal 148 CONTENTS Page

Nephrold 152 Parallel Curves 155 Pedal Curves 160 Pedal Equations 166 Pursu.t Curve 170 Radial Curves 172 Roulettes 175 Semi-Cubic Parabola 186 Sketching 188 Spirals 206 Strophoid 217 Tractrix 221 Tr!gonometric Functions 225 Trochoids 233 Witch of Agnesi 237 This volume proposes to supply to studentand teacher a quick reference on properties of planecurves. Rather, than a systematic or comprehensive studyof curve theory, It is a eoilectioi' of informationwhich might be found usefu7 in the classroom and in engineeringpractice. The alphabetical arrangement Is givento aid In the search for this information.

Tt seemed necessary to incorporatesections on such topics as Evolutes, Curve Sketching,and Intrinsic Equa- ticns to make the items and propertieslisted under various curves readily understandable, If the bog': is used as a text, it would be desirableto present the material in the following order:

I II ANALYSIS and SYSTEMS CURVES Caustics Astroid Curvature Cardioid Envelopes Cassiniun Curves Evolutes Catenary Functions with Discontinuous Circle Properties Cissold Clissettes Conchoid Instantaneous Centers Conics Intrinsic Equations Cubic Parabola Inversion Cycloid Involutes Deltoid Isoptic Curves Epi- and Oypocycloid Parallel Curves Exponential Curves Pedal Curves Folium of Descartes Pedal Equations Hyperbolic Functions Radial Curves Kieroid Roulettes Lemniscate Sketching Limacon Prochoide Nephrold Pursuit Curves Semi-cubic Parabola Spirals Strophoid Tractrix Trigonometric Functions Witch "77.'1-

PREFACE Since derivations of all propertieswould make the volume cumbersome and somewhatdevoid of general interest, explanations are frequentlyomitted. It is thought possible for the reader to supply manyof them without difficulty. Space is provided occasionally for thereader to in- sert notes, proofs, and references ofhis own and thus fit the material to his particular interests. It is with pleasure that theauthor acknowledges valuable assistance in the composition ofthis work. Mr. H. T. Guard criticized themanuscript and offered helpful suggestions; Mr. CharlesRoth and Mr. William Bobalke assisted in the preparationof the drawings; Mr. Thomas Vecchio lent expertclerical aid. Apprecia- tion is also due Colonel Harris Joneswho encouraged the project.

Robert C. Yates West Point, N. Y. June 1947

For an informative account see"Historical Stages in the Definition of Curves"by C. B. Boyer in National Mathematics Magazine, XIX(1944-5) 294-310. HISTORY: The Cycloidal curves,including the Astroid, were discovered by Roemer (1674)in his search for the best form for gear teeth. Double generation was first noticed by Daniel Bernoulliin 1725.

1. DESCRIPTION: The Aatroid is a hypocycloidof four cusps: The locus of a point Pon a circle rolling upon the inside of another with radius four timesas large.

(a) Pig. 1 (b) bauble Generation: it may also be described by a point on a circle of radius It rollingupon the inside ofsa fixed circle of radiusa. (See 8picycloids) 2 2. EQUATIONS: a oos3t =(i)(3cos t 4 cos 3t) i +yi. a y = a stk3t -1)(3 sin t sin 3t)

2 2 2 z:a - ,5p

7,a COS 2 s = 1.1

3. METRICAL PRoPERTTES:

L 6a

Vx =(51.)(ra3)

.11 . t

4. GENERAL tTEMS: (a) Its cvulute is another Astroid.[See Evolutes /1(b).] (h) It is the envelope of a family ofEllipses, the sum of whose axes isconstant, (Fig. 2b)

(a) ASTROID (c) The length of its tangent interceptedbetween the 'cusp tangents is constant. Thus it is the enveloa or a Trammel or Archimedes. (Fig. 2a) (d) Its orthoptic with respect to itscenter is the curve

a2 r2= (--2).cos2 20.

(e) Tangent Construction:(rig. 1) Through P draw the circle with center on theCircle of radals24 which is tangent to the fixed circle asat T (left-hand figure). Since the instantaneous center ofrotation of P is T, TP is normal to the curveat P.

BIBLIOGRAPHY

Edwards, J.: Calculus, Macmillan(1892) 337. Salmon, O.: Higher Plane Curves,Dublin (1879) 278. Wieleitner, H.: Speziefle ebeheKurven, Leipzig (1908). Williamson, B.: DifferentialCalculus, Longmans, Green (1895) 339. Section on Epicycloids, herein. CARDIOID

HISTORY: The Cardloid is a memberof the family of Cy- cloldal Curves, first studied byRoemer (1674) in an in-' vestigation for the best form ofgear teeth.

1. DESCRIPTION: The Cardloidis an Epicycloid of one cusp: the locus of a point P ofa circle rolling upon the outside of another ofequal size. (Fig. 3a)

(a) Fig. 3

Double Generation: (Fig. 3b). Let thecurve be generated by the point P on the rolling circleof radius a. Draw ET', OT1F, and PT' to T.Draw FP to D and the circle through T, P, D. Since angle DPTm t, this last circle has DT as diameter. Now, PD is parallelto TIE and from similar triangles, DEm 2a. Moreover, arc TT' m au = arc T'P = arc T'X. Accordingly,

arc TTIX = 2a0 m arc PP. CARDIOID 5

Thus the curve may bedescribed as an Epicycloidin two ways: by a circleof radius a, or by one ofradius 2a, rolling as shown upon afixed circle of radius a.

2. EQUATIONS:

(x2 + y2 + 2ax)2 = 4a2(x2 +Y2)(Origin at cusp). cusp). r = 2a(1 + cos0), r = Pa(1 + sin 0)(Origin at 9(r2 - a2) = 8p2. (Origin atcenter of fixedcircle). 2t) x = a(2 cos t - cos esit). z = a (2eit- ' y = a(2 sin t - sin2t) ,,, [ s =8a'cos(). r3 = 4ap2. 3 9R2 + s2 . 64a2.

3. METRICALPROPERTIES: L = 16a. A = 6na2 (10)(ne2) Ex US =(i)t 5

R Wtar for r = a(1 - cos 0).

4. GENERAL ITEMS: (a) It is the inverse of aparabola with respect to its focus.

(b)Its evolute is anothercardioid. respect to a point on the circle. (d) It is a special limacon: r = a +b cos 0 with a se (e) It is tAe caustic of .a circlewith radiant point on the circle. (f) The tangents at the threepoints whose subtended angles, measured at the cusp,differ by

2Sare parallel. four (g) The 29g of distances from the cusp to the intersections with an arbitraryline is constant,

J C ARD IOID

(h) Cam. rr the cardioldbe pivoted at the cusp and rotated with constant angularvelocity, a pin, con- traired to a i stra:!-ht and bearin on the jardiold, will move with simple harmonicmotion. Thus Cor

= -(a sin 0)0, = -(a cos 062 - (a sin 06. IV 6 = k, a constant:

= -k2(a cos u) = -k2(r a), or

d2, --pkr - a) = -k 2,kr dt - a),

the d:fierential equationcharacterizing the motion or any point of the pin.

/0) A' C)I

Fig. )1

(i) The curve Is the locusof the point P of two --similar (Proportional) crossedparallelograms, joined ac shown, with points 0 and A fixed.

L. --'"7777.1r7.7r7,71

CARDIOID 7 AB = OD = b; AO .BD = CP = a; BP = DC . c=b + 2a. 2 .:tnd a = bc. At all times, angle PCO = U = angle CuX. Any point rigidly attached to CP describes a Limacon.

BIBLIOGRAPHY

Keown and Faires: Mechanism, McGraw Hill(1931). Morley and Morley: Inversive Geometry, Ginn(1933) 239. Yates, R.C.: Geometrical Tools, EducationalPublishers, St. Louis (1949) 182. CASSINIAN CURVES

HISTORY: Studied by Giovanni Domenico Cassini in 1680 in connection with the relative motions of earth andsun.

1. DESCRIPTION: A Cassinian Curve isthe locus of a point P the produtt of whose distances fromtwo fixed points Fi, F2 is constant (here= k2).

/ Y % 1 I I I i o

Fig. 6

2. EQUATIONS : [(x - a)'+ y2J [(x +a)2 + y2 j r4 + a4- 2r2a2 cos 20 = k4. (-a,o) F2 = (aon

3. METRICAL PROPERTIES: (See Section on Lemniscate) C ASSINIAN CURVES 4. GENERAL ITEMS: (a) Let b - a be the inner radius 07 a torus whose generating circle has radius a. The section formed by a plane Parallel to the axis of the torus and distant a units from it is a Cassinian. If b = 2a this plane is an inner tangent to the surface and the section is a Lemniscate. (b) The set of Cassinian Curves (x2 + y2)2 + A(y2 - x2) +B= 0, S pi 0, inverts into itself. (c) If k = a, the Cassinian is the LemnIscate of Bernoulli: r2 = 2a7cos 20, a curve that is the inverse Fig. 7 and 2edal, with respect to its center, of a Rectangular Hyperbola. (d) The points P and P' of the linkage shown in Fig. 8 describe the curve. Here AD = AO = OB = a; DC = CQ = E0 = OC CP = PE = EP' = P'C = d. 2 '

Fig. 8 10 CASSINIAN CURVES Let the coordinates and P be (p,u) and (r,e), respcLivol::. c, D, and lie o!: a circle with 1k: ..(4 are always at right angles, Thus.

= (D0)2 -(Ft)2 uv p2= (!2 4a2 sin20. The attached Peatvllier ei1 Inverts the point Q to P under the property

tqr. d` .

Thus, ,!-1.1m1hatlhr.p :etweet. the lavttwo relations:

- -2)2 4 r!2- 4r2a:2s:n2U.

or, in rectah,.ular coord:nates:

(x2 + y212 + Ax2+ By'+ C = 0, a !urve that may 1)0 Identified asa Cassinian if d 4

(e) The 1us or the flex points ora family of confocal CassInlan curves is a Lemniscate of Bernoulli

5. POINTWISE CONSTRUCTUA:

Let the foci, 9, be F1, F2; the constant product k2. Lay off F1C = k perpendicular to F1F2. Draw the circle with center F1 and any radius FIX. Draw CX and Lts perpen- Fig. 9 dicular CY. Then

(Flx).(F,Y)=k2 CASSINIAN CURVES 11 and thus FIX and FlY are focal radii (measured from El and F2) of a point P on the curve. (From symmetry, four points are constructible from these two radii.) M is the midpoint of FIFO and A and B are extreme points of the curves.

BIBLIOGRAPHY

Salmon, G.: Higher Plane Curves, Dublin (1879) 44,126. Willson, P. N.: Graphics, Graphics Press (1909) 74. Williamson, B.: Calculus, Longmans, Green (1895) 233,333. Yates, R.C.: Geometrical Tools, Educational Publishers, St. Louis (1949) 186. CATENARY

HISTORY: Galileo was the first to investigate the Cate- nary which he mistook for a Parabola, James Bernoulli in 1691 obtained its true form and gave some of its properties,

1. DESCRIPTION: The Catenary is the form assumedby a perfectly flexible inextensible chain of uniform density hanging from two supports not in the same vertical line.

*if

N L.- KI

A x

Fig. 10

2. EQUATIONS: If T is the tension at any point P,

P cos' cp a s= ay' =a tan p p aR = s T sin Y = ks

a a a% 2 2 y a.cosh(!)= H(e ) ; Y a a 2 CATENARY 13 3. METRICAL PROPERTIES: A = ass = 2(area triangle PCB) Ex= n(YS ax)

R Vx = (2)11x

N = -R.

4. GENERAL ITEMS: (a) The tangent at any point (x,y) is also tangent to a circle of radius a, with center at(x,0).

=sinh(A) 1/2 a2 . a - a

(b) Tangents drawn to the curves y =ea, y = e a, y = a cosh at points having the same abscissa are a concurrent. (c) The path of B, an involute of the catenary, is the Tractrix. (Since tan U = PB = s). al (d) As a roulette, it is the locus of the focus of a parabola rolling along a line. (See Envelope3, 5g) (e) It is a plane section of the surface of leaat area (a soap film catenoid) spanning two circular disks, Pig. ila. (This is the only minimal surfaceof reVelu- tion.)

(a) Pig, 11 (b) 14 CATENARY

(f)it is a plane section of a sail bounded by two rods with the wind perpendicular to the plane of the rods, such that the pressure on any element of the sail is normal to the element and proportional to the square of the velocity, Fig. lib. (See Routh)

BIBLIOGRAPHY

EncYclonaedia Britannica, 14th Ed. under "Curves, Special". Routh, E. J.: Analytical Statics, 2nd. Ed. (1896) I 0458, P. 310. Salmon, G.: Higher Plane Curves, Dublin (1879) 287. Wallis: Edinburgh Trans. XIV, 625. CAUSTICS

HISTORY: Caustics were First introduced and studied by Tschirnhausen in 2682. Other contributors were Huygens, Quetelet, Lagrange, and Cayley.

1. A caustic curve is the envelope or light rays, emitted from a radiant point source S, after reflection or refraction by a given curve f = 0. The caustics by reflentton and refraction are called cataeaustic and diacaus- tic, respectively.

2. An orthotomic curve (or secondary caustic) is the locus of the point t, the reflection of S in the tangent at T. (See also Pedal Curves.)

3. The instantaneous center of motion of is T. Thus the caustic is the envelope of normals, TQ, to the craw- to_mtpi i.e., the caustic is the evolute of the orthe- t0MiP.

4. The locus of P is the pedal of the reflecting curve with respect to S. Thus the orthotomic is a curve SIMI.- lard to the pedal with double its linear dimensions. 16 CAUSTICS 5, The Cataeaustic oe a circle is the evolute of a limacon whose pole is the radiant point. With usual x,y axes tradius a, radiant point (c,U)), the equation of the caustic is:

(( 402a2)(x2+ y2) -2aacx-a2c2) 327a4c2y2(x2+ y2..c2)2 sO. For various radiant points C, these exhibit the following forms:

(a) (b)

(0)

(e) Fig. 1, (f) CAUSTICS 17 6. In two particular cases, the caustics of a circle of radius a may be determined in the following elementary way:

(a) Fig. (b)

With the source S at et, With the source S on the the incident and reflected circle, the incident and rays make angles 0 with reflected rays makes angles the normal at T. Thus the 0/2 with the normal at T. fixed circle 0(A) of Thus the fixed circle and radius a/2 has its arc AB the equal rolling circle equal to the tv,0 AP of the have arcs AB and AP equal. circle through A, P, T of The point P generates radius a/4. The point P of Cardioid and TPQ is its tan- this latter circle gener- gent (AP is perpendicular ate. the Nephroid and the to TP). reflected ray TPQ is its tangent (AP is perpendicular to TP). These are the bright curves seen on the surface of cof- fee in a cup or upon the table inside of a napkin ring. 18 CAUSTICS 7. The Caustics by Refraction (Diacaustics) at a Line L. SQ 13 Incident, QT refracted, and S is the reflection of S In L. Prvdc.co TQ to rtltA Lhu variable circle drawn through S, Q, and S in P. Let the angles of incidence sin01 and refraction be 01 and02and p. ------be the index sin 02 of refraction. Now SP and -§P make equal angles with the refracted ray PQT. Thus in passing from a dense to a rare medium (01 < 02) and vice versa (U1 > 02):

92 F. S

L RARE DENSE

0, > 0, ./I >

Fig. 15

sin 0 AS sin 0 PS PS CAUSTICS 19 AS 4, AE SE - AS SE u PS 4. PS PS + PS PS - PS PE - PS Thus, since SE is ,wistar,t, Thus, since SS Is constant,

PS 4, = SE/14 - PS = SS /!t a constant. The locus of Pa constant. The locus of P is then an ell:pse with S, is then an hyperbola wSth S as foci, major axIs SS/i4, S, S as foci, major axis eccentricity 14,ar:d with Sg/4, eccentricity V., and PQT as its normal. The with PQT as its normal. The envelope of these rays PQT, envelope these rays PQT, normal, to the ellipse, is normal to the hyperbola is its evolute, the caustic. its evolute, the caustic. (Fig. 16) (Fig. 17)

F. 16 Pig. 17

8. SOME EXAMPLES: (a) If the radiant point is the focus of a parabola, the caustic of the °volute of that parabola is the evolute of another parabola. 20 CAUSTICS (b) If the radiant point is at the vertex of a reflecting parabola, the caustic is the evolute of a cissoid. (c) If the radiant point is the center of a circle, the caustic of the involute of that circle is the evolute of the spiral of Archimedes. (d) If the radiant point is the center of a conic, the reflected rays are all normal to the quartic 2 r = A cos 20 + B, having the radiant point as double point. (e) If the radiant point moves along a fixed diameter of a reflecting circle of radius a, the two Cusps of the caustic which do not lie on that diameter move on 0, the curve r = acom.-).

(f) If the radiant point is the pole of the reflecting spiral r = aee eta a, the caustic is a similar Spiral,. (g) If light rays parallel to the y-axis fall upon the reflecting curve y = ex, the caustic is a catenary. (h) The orthotomic of a parabola for rays perpendicular to its axis is the sinusoidal spiral r = a.sec 3().

BIBLIOGRAPHY

Aperican Mathemops1 28(1921) 182,187. Cayley, A.: "Memoir on Caustics ", Philosophical Trans- actions (1856). Heath, R. S.: Oo6mqI1194 WI:pi (1895) 105. Salmon, G.:Hig'lane Hpr 'lane qung, Lublin(1879) 98. THE CIRCLE

1. DESCRIPTION: A circle Is a plane continuous curve all of whose points are equidistant from a fixed coplanar point.

2. EQUATIONS: k)2 a2 (x - h)2 + (Y- [x = h + a cos 0 2 2 X 4. y + Ax + By + C = 0 y = k + a sin 0

2 2 X -I. y x y 1 2 =0 s = ap XI + y.2 XI Y1 1 2 2 R = a X2 1 + Y2 x2 Y2 r2 2 2 X3 -I. y3 x3 y3 1

3. METRICAL PROPERTIES: L = 2na E = 4na2 R = a

A = na2 V =

4. GENERAL ITEMS: (a) The Secant Property. Fig. 18(a). If lines are drawn from a fixed point P intersecting a fixed circle, the product of the segments in which the circle divides each line is constant; i.e., PA.PB = ppPC (since the arc subtended by L BCD plus that subtended by L BAP is the entire circumference, triangles PAD and PBC are similar). To evaluate this constant, p, draw the line through P and the center 0 of the circle. Then (PO - a)(P0 + a) = p = (P0)2 - a2. The quantity p is called the power of the point P with respect to the circle. If p (, = > 0, P lies respectively inside, on, outside the circle. 22

(n) Fig. 18 (b)

The locus of all points P which have equal power with respect to two fixed circles Is a line called the radical axis or the two circles.It' the circles intersect, the radical axis is their courier' chord. Fig. 18(h). The three radical axes of three circles intersect in a point called the radical center, a point having equal power with re3pect to each or the circles and equidistant from them. Thus to construct the radical axis of two circles, first draw a third arbitrary circle to intersect the two. Common chords meet on the required axis. (b) Similitude. Any two coplanar circles have centers of similitude; the Intersections I and E (collinear with the centers) of lines Joining extremities of parallel diameters. The six centers of similitude of three circles lie by threes on four straight lines. The excenter of similitude of the circumcircle and nine-point circle of a triangle is Its orthocenter. THE CIRCLE

Fig. 19

(c) The Problem of Apollonius is that of con- structing a circle tangent to three given non-coaxal circles (generally eight solutions). The problem is reducible (see Inver- sion) to that of drawing a circle through three specified points.

Fig. 20 THE CMCLE 24 N (d) Trains. A series of circles each drawn tangent to two given non-intersecting circles and to another mem- 5ov of the sevieo is called a train. It is not to be

Fig. 21

ekpect.3d that such a series generally will close upon itselv.If such is the case, however, the series is called a Steiner chain. Ally Steiner chairs can be inverted into a Steiner chain tangent to two concentric circles. Two concentric circles admit a Steiner chain of n circles, encircling the common center k times if the angle subtended at the center by each circle of the train is commensurable with 360°, i.e., equal to

(:)(3600).

If tvo circles admit a Steiner chain, they admit an infinitude of such chains. THE CIRCLE 25 (e) Arbelos. The figuft bounded by the semicircular arcs AXB, BYC, AZC (A,B,C collinear) is the arbelos or shoemaker's knife. Studied by Archi- medes, some of its properties are: 1. AB = A. 2. Its area equals Fig. 22 the area of the circle on BZ as a diameter. 3. Circles inscribed in the three-sided figures ABZ, CBZ are equal with diameter AC 4. (Pappus) Consider a train of circles co, ol, 02, ... all tangent to the circles on AC and AB (co is the circle BC). If rn is the radius of cn, and hn the distance from its center to ABC,

(Invert, using A as center.)

BIBLIOGRAPHY

Daus, P. H.: College Geometry, Prentice-Hall (1941). Johnson, R. A.: Modern Geometry, Houghton Mifflin (4929) 113. Mackay, J. S.: Proc. Ed. Math. Soc. III (1884) 2. Shively, L. S.: Modern Geometry, John Wiley (1939) 151. CISSOID

HISTORY: Diocles (between 250-100 BC) utilized the ordinary Cissoid (a word from the Greekmeaning "ivy") in finding two mean proportionals between given lengths a,b (i.e., finding x such that a, ax, axe, b form a geometric progression. This is the cuberoot problem since

b, x =). Generalizations follow. Asearly as 1689, a J. C. Sturm, in his Mathesis Enucleata, gave amechanical device for the construction of the Cissoid of Diocles.

1. DESCRIPTION: Given two curves y = fi(x), y ==f2(x) and the fixed point O. Let Q and R be the intersections of a variable line through 0 with the given curves. The locus of P on this secant such that OP m (OR) - (OQ) = QR is the Cissoid of the two curves with respect to O. If the two curves are a line and a circle, the Fig. 23 ordinary family of Cissoids is generated. The discussion following is restricted to this family. Let the two given curves be a fixedcircle of radius a, center at K and passingthrough 0, and the line L perpendicular to OX at 2(a + b) distance from 0.The ordinary Cissoid is the locus of P on thevariable secant through 0 such that OP = r = QR. The generation may be effected by theintersection P of the secant OR and the circle of radius atangent to L at R as this circle rolls upon L.(Pig. 24) CISSOID 27 The curve has a cusp if b = 0 (the Cis- sold of Mucles); a double point if the rolling circle passes between 0 and K. Its asymptote is the line L.

Nri

Fig. 24

2. EQUATIONS: x2(2b- x) r = 2(a + b)sec 0 - 2a cos H. Y2 [x-2(a+b)) 2[b + (a + b)t21 x= (1 + t2) 2.(bt + (a + b)t3) Y = (1 + t2) 3 x b = 0: r = 2a. sin 0 tan 0; Y2 - % . the = (2a -x) Cissoid of Diocles),

3. METRICAL PROPERTIES: Cissoid of Dlocles: V(rev. about asymp.) = 2n2a3

1(area betty, curve and asymp.) 3 A(area betw. curve and asymp.)=ne2 28 C1SSOID 4. GENERAL ITEMS: (a) A family of these Cissoids may be generated by the Peaucellier cell arrangement shown. Since (0Q)(QP) = le = 1, 2c.cos 0(2c.cos 8 + r) 1 or

1 r = (--)sec 0 - 2cscos 8, 2c 1 which, for c < - > 2 has, respectively, no loop, a cusp, a loop.

Fig. 25

(b) The Inverse of the family in (a) is, if r'p - 1, (center of inversion at 0) y2 + x2(1 - 4c2) = 2cx, 1 an Ellipse, a Parabola, an Hyperbola if c< = > , respectively. (See Conics, 17). (c) Cissoids may A be generated by the carpenters Igare with right angle at Q (Newton). The fixed point A of the square moves along CA while the other :"4-Q edge of the square passes through bp a fixed point on the line BO perpendicular to

Fig. 26 MCC 29 AC. The path of P, a fixed point cn AO describes the curve. Let AP = OB = b, and BC = AO = 2a, with 0 the origin of coordinates. Then AB = 2a. sec 0 and

r = 2a. sec 0- 2`). cos 0. The point Q describes a Strophoid (See Strophoid 5e).

(d) Tangent Construction: (See Fig. 26) A has the direction of the line CA while the point of the square at B moves in the direction BQ. Normals to AC and BQ at A and B respectively meet in H the center of rotation. HP is thus normal to the path of P.

x3 (e) The Cissold y2 = is the pedal of the Parabola y2 = -4ax with respect to its vertex.

(f) It is a special Kieroid.

(g) The Cisso-d as a roulette: One of the curves is the locus of the vertex of a parabola which rolls upon an equal fixed one. The common tangent reflects the fixed vertex into the position of the moving vertex. The locus is thus a curve similar to the pedal with respect to the vertex.

(h) The Cissoid of an algebraic curve and a line is itself 12111)1112. 30 MEWED (I) The Cissoid cr a line and a circle with respect to Its center is the Con.2hoid of Nicomedeq.

(j) The Strophold Is the Cissoid of a circle and a 1 ne through Its center with respect to a point of the c:-cle. The Cissold of Dlocles is used in the design or planing hulls (See Lord).

(k) The Cissoid of 2 concentric circles with respect to their center is a circle.

(1) The Cissoid of a pair of parallel lines is a line.

BIBLIOGRAPHY

Hilton, H.: Plane Algebraic Curves, Oxford (1932) 175, 203. Wieleitner, H.: Spezielle ebene Kur,en,LeipsIg (1908) 37ff. Salmon, G.: Hig!her Plane Curves, Dublin (1879) 182ff. Niewenglowski, B.: Cours de G4omettrie AnaVtique, Paris (1895) II, 115. Lord, Lindsay: The Naval. Architecture or 113_22.111D1 Hulls, Cornell Maritime Press (1946) 77. CONCHOID

HISTORY: Nicomedes (about 225 BC) utilized the Conchoid (from the Greek meaning "shell-like") in finding two mean proportionals between two given lengths (the cube-root , problem).

1. DESCRIPTION: Given a curve and a fixed point O. Points PI and Pg are tak.3n on a variable line through 0 at distances + k from the intersection of the line and 2 curve. The locus of PI and P20 is the Conchoid of the given 0-- curve with respect to O.

Fig, 27

The Conchoid of Nic-Atedes is the Conchoid of a Line.

Fig. 28

The Limgcon of Pascal Is a Conchoid of a circle, with the fixed point upon the circle. 32 CONCHOID 2. EQUATIONS: General: Let the given curve be r = f(0) and 0 be the origin. The Conchoid is

r = f(Q) + k. The Conchoid of Nicomedes (for the figure above) is:

r = a.csc 0 + k, (x2 y2)(y a)2 k2y2, which has a double point, a cusp, or an isolated point if a < = > k, respectively.

3. METRICAL PROPERTIES:

4. GENERAL ITEMS:

(a) Tangent Construction. (See Fig. 28).The perpendicular to AX at A meets the perpendicular to OA at 0 in the point H, the center of rotation of any point of OA. Accordingly, HP/ and HP2 are normals to the curve. CONCHOID 33 (b) The Trisection of c.n Angle XOY by the marked ruler involves the Conchoid of Nicomedes. Let P and Q be the two ma:kr: or. the ruler 2k units apart. Construct BC parallel to OX such that OB = k. Draw BA perpendicular to BC. Let P move along AB while the edge of the ruler passes through 0. The point Q traces a Conchoid and when this point falls on BC the angle Is trisected.

Fig. 29

BIBLIOGRAPHY

Mortiz, R. E.: Univ. of Washington Publications,(1923) (tor Conchoids of r = cos(P/00]. Hilton) H.: Plane Algebraic Curves, Oxford(1932). Yates, R. C.: The Trisection Problem, West Point, N. Y. (1942) CONES

1. DESCRIPTION: A cone is a ruled surfaceall of whose line elements pass through a fixed pointcalled the vertex.

2. EQUATIONS: Given two surfacesf(x,y,z) = 0, g(x,y,z) = 0. The cone through their common curve with vertex V at (a,b,c) is found as follows. Let PI:(xlal,z1) be on the given curve and P:(x,y,z) a point on the cone which lies collinear with V and Pi. Then

i,- a =k(xl a), y - b = k(yi - b), z - c = k(zi -c), for all values of k. Thus the curve f(x1,5,1,z1) = 0 g(xial,z1) = 0 Fig. 30 pro1 uces the cone:

fc(x -a) (y -b) (z - c) 4. a' 4. b, 4. cl =0 k k k (x - a) (y - b) lEIE/ =0 gf k + a' k + 1), k + c] [ Since this condition must exist forell values k, the elimination of k yields the rectangularequation of the cone.

*Thus any equation homogeneous inxpytzis a cone with vertex at the CONES 35 3. EXAMPLES: The cone with vertex at the origin containing the curve x2 32 x2 + y2- 2z I 30 is or x2 + y2 - 222 =0. z 1 . 0 Z - k 0

The cone with vertex at the origin containing the curve

x2-22+ 32-143 =0 I2-21sx+ 32- 4ky .0 2y 2 3 2 is or 2x -xz +2y-2yz =0. z2 wry Z2-4y=0

Thecone with vertexat (1,2,3) containing the curve

=0 f (x-1)2k+ (y-2)21 2( x-1) + 4(y -2)) 2( z-3) - 1 0 e k k- in

z -4 0 1st 0

or(x1)2+ (y-2)2+2(x-1)(z -3)+4(y -2)(z -3)- 3(z-3)2 0.

BIBLIOGRAPHY

Smith, Gale, Neelley:Analytic Geometry, Ginn(1938)284. CONICS

HISTORY: The Conics seem to have been discovered by Menaechmus (a Greek,c.375-325 BC), tutor to Alexander the Great. They were apparently conceived in an attempt to solve tr,a three famous problems of trisecting the angle, duplicating thg cube, and eauaring the circle. Instead of cutting a single fixed cone with a variable plane, Menaechmus took a fixed intersecting plane and cones of varying vertex angle, obtaining from those having angles < = > 90° the Ellipse, Parabola, and Hyperbola respectively. Apollonius is credited with the definition of the plane locus given first below. The ingenious Pascal announced his remarkable theorem on inscribed hexagons in 1639 before the age of 16.

1. DESCRIPTION: A Conic is the locus of a point which moves so that the ratio of its distance from a fixed point (the focus) divided by its distance from a fixed line (the directrix) is a constant (the eccentricity e), all motion in the plane of focus and directrix (Apol-

lonius). If e< =,> 1, the locus is an Ellipse, a Parabola, an Hyperbola respectively.

Fig. 31

2kx +kg ek 72+ (1.02)? , r 0- (i* e ein 0) (3. t COO0) CONICS 37 2. SECTIONS OF A CONE: Consider the right circular cone of angle p cut by a plane APFD which makes an angle m with the base of the cone. Let P be an arbitrarily chosen point upon their curve of intersection and let a sphere be inscribed to the cone touching the cutting plane at F. The element through P touches the sphere at B. Then PF = PB. Let ACBD be the plane containing the circle of inter? section of cone and sphere. Then if PC is perpendicular to this plane, PC =(PA)sina = (PB)sin 0 me (PF)sin 0, or

PF sins Fig. 32 (PA sin 0'I e, a constant as P varies (01,I3 constant) . The curve of intersection is thus a conic according to the definition of Apollonius. A focus and corresponding direotrix are F and AD, the intersection of the two planes. NOTE: It is evident now that the three types of conics may be had in either of two ways: (A) By fixing the cone and varying the intersecting plane (0 constant and a arbitrary); or (B) By fixing the plane and varying the right circular cone (a constant and 0 arbitrary). With either choice, the intersecting curve is

an Ellipse if a < 0, a Parabola ifamo, an Iblierbola if a > 0 38 CONICS 3. PARTICULAR TYPE DEMONSTRATIONS:

ProPe,pro pe er,-00,t Aft,a Constant

Fig. 33 CONICS 39 It seems truly remarkable that these spheres, inscribed to the cone and its cutting plane, should touch this plane at the foci of the conic - and that the directrices are the intersections of cutting plane and plane of the intersection of cone and sphere.

4. THE DISCRIMINANT: Consider the general equation of the Conic: Axe + 2Bxy + Cy2 + 2Dx + 2Ey + F .2 0 and the family of lines y = mx.

Figs 34

This family meets the conic in points whose abscissas are given by the form:

(A 2Bm + Cm2)x2 + 2(D + Em)x + F sa 0. 4o CONICS If there are lines of the family which out the curve in one and only one point,* then

B */B2-AC A + 2Bm + Cm2 = 0 or m =

The Parabola is the conic for which on1:4 one line of the family cuts the curve just once. That is, for which: B2- AC = O. The Hyperbola is the conic for which two and only tvo lines cut the curve just once. That is, for which: B2 - AC > O. The Ellim is the conic for which no line of the family cuts the curve just once. That is, for which: B2- AC < O.

5. OPTICAL PROPERTY: A simple demonstration of this outstanding feature of the Conics is given here in the case of the Ellipse. Similar treatments may be presented for the Hyperbola and Parabola. The locus of points P for which FLP + F2P 2a, a constant, is an Ellipse. Let the tangent to the curve be drawn at P. Now P is the only point of the tangent line for which FiP + PaP is a minimum. For, consider any other point Q. Then FiQ + FaQ > FiR + F2R 2a - FiP+ PgP, But if FLP + FRP is a minimum, P must be collinear with FL and Pie, the Fig. 35 reflection of Fa in the T A point of tangency he is counted algebraically as two points, the "point at 0" is excluded. CONICS 41 tangent. Accordingly, since a = 0, the tangent bisects the angle formed ny the focal radii.

o. POLES AND POLARS: Consider the Conic: Axe + 2Bxy + Cy 2 + 2Dy + 2Ey + F = 0 and the point P:(h,k) The line (whose equation has the form of a tangent P: (1,111 to the conic): Ahx + B(hy + kx) + Cky + D(x + h) + E(y + k) + F = 0 (1) (At.Y2) is the polar of P with respect to the conic and P is its pole. Let tangents be drawn from P to the curve, meet- Fig. 36 ing it in (xIal) and (k21y2), Their equations are satisfied by (h,k) thus: Ahxi + B(hyl+kxx) + Ckxi + D(xx + h) + E(yx +k) + F 0

Ahx2 + B(hy2+kx2) + Ckx2 + D(x2+k) + E(y2 +k) + F 0. Evidently, the polar given by (1) contains these points of tangency since its equation reduces to these identi ties on replacing x,y by either xial or x2a2. Thus, if P is a point from which tangents may be drawn, its polar is their chord of contact, Let (a,b) be a point on the polar of P. Then Aha + B(hb +.ka) + + F = O. This expresses also the condition that the polar of (a,b) passes through (h,k). Thus 42 CONICS Tr P lies on the polar of g, then S lies on the polar of P. Ir. other words, .1f a point move on a fixed line, its polar passes through a fixed point, and conversely. Note that the location of P relative to the conic does not affect the reality of its polar. Note also that if P lies on the conic, its polar is the tangent at P.

7. HARMONIC SECTION: Let the line through P2 meet the conic in Ci1,Q2 and its polar in Pi. These four points form an harmonic set and

iPIQ1) (Q2P1 i.e., Q1 (Q02) (Q2P2 and Q2 divide the segment PIP2 internally and extern- ally in the same ratio, and vice versa. In other words,. given the conic and a fixed point P2: A variable line through P2 meets the conic in Ci1,Q2. The locus of Pi which, with P2, divides Q1Q2 harmonically is the Fig. 37 polar of P2.

The segments P2Q1, P2Pi, P2Q2 are in harmonic progression. That is:

2 1 1 (POI) (PAO+7172 0 CONICS 43 S. THE POLAR OF P PASSES THROUGH R AND S, THE INTERSEC- TIONS OF THE CROSS-JOINS OF SECANTS THROUGH P. (Fig. 38a)

(a) Fig. 38 (b)

Let the two arbitrary secants be axes of reference (not necessarily rectangular) and let the conic (Fig. 38b) 2 2 Ax 2Bxy .1. Cy 2Dx 2Ey F = 0 have intercepts al,a2; bl,b2 given as the roots of Ax2 2Dx F = 0 and Cy2 2Ey F = O. Prom these

1 2D or D = (-E)(1- al a2 2 al a2

1 .1 2g. = - or E (- )(T)1: bl be Nov the polar of P(0,0) is Dx + Ey + F = 0 or

1 1 1 1 x(-- --) y(-- --) - 2 = O. al a2 b2 44 CONICS The cross-joins are:

+ . and 25.- + . 1 al D a a2 b l The family of lines through their intersection R:

A. jL 1 + x(A. JL 1) 0. al b2 a2 bl contains, for , m 1, the polar of P.Accordingly, the polar of P passes through R, and by inference, through S.

This affords a simple and classical construction by the straightedge alone of the tangents to a conic from a point Ps

Draw arbitrary secants from P and, by the intersections of their cross- joins, establish the polar of P. This polar meets the conic in the points of tangency, 9. PASCAL'S THEOREM: One of the most far reaching and productive theorems in all of geometry is concerned with hexagons inscribed to conics. Let the vertices of the hexagon be numbered arbitrarily* 1, 2, 3, 1', 2', 3'. The intersections X, Y, Z of the loins (1,2'0112)

(1,31;11,3) (2,3';2',3) are collinear, and conversely. Apparently simple in character, it nevertheless has over 400 corollaries important to the structure of synthetic geometry. Several of these follow.

Fig. 40

By renumbering, many such Pascal. lines correspond to a single inscribed hexagon, 46 CONICS 10. POINTWISE CONSTRUCTION OF A CONIC DETERMINED BY FIVE GIVEN POINTS: Let the five points be numbered 1,2,3,11,2'. Draw an arbitrary line through 1 which would meet the conic in the required point 3'. Establish the two points Y,Z and the Pascal line. This meets 2'3 in X and

0finally 2,X meets the 0 arbitrary line through 1

in 3'. Further points are located in the same way. Fig. 41

11. CONSTRUCTION OF TANGENTS TO A CONIC GIVEN ONLY BY' FIVE POINTS: In labelling the points, consider 1 and 3' as having merged so that the line 1,3' is the tangent. Points X, 2 are determined and th. Pascal line drawn to meet 11,3 in Y. The line from Y to the point 1=3' is the required tangent. The tangent at any other point, determined as in (10), is constructed in like fashion.

Fig. 42 CONICS 47 12. INSCRIBED QUADRILATERALS: The pairs of tangents at opposite vertices, together with the opposite sides, of quadrilaterals inscribed to a conic meet in four collinear points. This is recognized as a special case of the inscribed hexagon theorem of Pascal.

Fig. 43

13. INSCRIBED TRIANGLES: Further restriction on the Pascal hexagon produces a theorem on inscribed triangles. For such triangles, the tangents at the vertices meet their opposite sides in three collinear points.

Fig. 44 48 CONICS

14. AEROPLANE DESIGN: The construction of elliptical sections at right angles to the center line of a fuselage is essentially as follows. Construct the conic given three points Pi, P2, P3 and the tangents at two of them. To ob- tain other points Q on the conic* draw an arbitrary Pascal line through X, the intersection of the given tangents, meeting P1P2 in Y; P1P3 in Z. Then YP3 and ZP2 meet in Q. Fig. 45

15. DUALITY: The Principle of Duality, inherently fundamental in the theory of Pro- jective Geometry, affords a corresponding theorem for each of the foregoing. Pascal's Theorem (1639) dualizes into the theorem of Brianchon (1806): If a hexagon circumscribe a conic, the three joins of the opposf vertices are concurrent. (This is apparent on polarizing the Pascal hexagon.)

Fig. 46 CONICS 49 16, CONSTRUCTION AND GENERATION; (Seealso Sketching 2.) The following are a few selected frommany. Explanations are given only where necessary. .1 4 (a) String Methods:

El I. Iph,. Hyperbola

Fig. 47

(b) Point-wise Construction:

f. cog i Q 1* Site I t 0-1111*Ygd(at-11)1 ys b ton t

Fig. 48 50 CONICS (c) Two Envelopes:

(i) A ray is drawn from the fixed point F to the fixed circle or line. At this point of intersection a

line is drawn perpendicular to the ray. The envelope of this latter line is a conic*(See Pedals.)

(ii) The fixed point F of a sheet of paper is folded over upon the fixed circle or line. The crease

so formed envelopes a conic.(See Envelopes.) (Use wax paper.) (Note that i and ii areequivalent.)

* This is a Oliesettes the envelope of one Side of a Carpenter's square Whose corner moves along a circle While itsother leg passes through a fixed point. See Cissoid 4, and aliesettee 60. CONICS 51 (d) Newton's Method: Based upon the idea of twopro- jective pencils, the followin:!: is due to Newton. Two angles of constant magnitudes have vertices fixed at A and B. A point of intersection P of two of their sides moves along a fixed line. The point of intersection Q of their other two sides describes a conic through A and B.

Fig. 51

17. LINKAGE DESCRIPTION: The following is , selected from a variety of such mechanisms (see TOOLS). For the 3-bar linkage shown, forming a variable trapezoid: AB =CD =2a; AC BD= 2b; a > b;

(AD)(BC)=4(a2- b2). A point P of CD is selected and OP = r drawn parallel to AD Fig. 52 and BC. OP will remain parallel to these lines and so 0 is a fixed point. Let OM = c, MT = z, where M is the midpoint of AB. Then

L CONICS Al) = 2(AT)cos 0 = 2(a + 2)cos 0, 73,7 = 2(BT)cos 0 = 2(a - v)oos 0. Their product produces: 21 2 .2 (a2a - z !cos = 424 -b2. Combining this with r = 2(c + z)cos 0 there results

- c.cos0)2 = b2 - a2sin20 2 as the polar equation of the path of P. Inrectangular coordinates these curves are degenerate sextice, each composed of a circle and a curve resembling the figure W. If now an inversor OEPFP' be attacoed as shown in Fig, 53 so that r.p = 2k, where p = OP',

Pig. 53

the inverse of this set of curves (the locus ofP') is: b2 a2.p2,sin20, (k - c.p.cos 0)2 or, in rectangular coordinates: (a2 - b2)y7 - (b2 - c2)x2 2c.k.x +k2 m 0 a conic. Since a > b, the type depends uponthe relative value of cj that is, upon the position of theselected point P: CONICS 53 An Ellipse if c > b,

A Parabola if c t),

An Hyperbola if c < b.

(For an alternate linkage,see Cissoid, 4.)

18. RADIUS OF CURVATURE: For any curve in rectangular coordinates,

and N2= y2(1 + y12). Thus

IRI =427 . Y Y The conic y2 = 2Ax + Bx2, B > -1, where A is the semi-latus rectum, is an Ellipse if B < 0, a Parabola if B = 0, an Hyperbola if B > O. Here yy' m A + Bx, yy" + y'2 = B, andy3y" ysy,e Bye. Thus y3y" By2- (A +Bx)2 = -A2 and IRI 111A32

19. PROJECTION OF NORMAL LENGTH UPON A FOCAL RADIUS: Consider the conics pi(1 - e cos 0)= A, (A = semi-latus rectum).

Fig. 54 54 CONICS Since the normal at P bisects the anglebetween the focal radii, we have for the central conics:

-1PSF=-P-g FIQ pi or, adding 1 to each sideof the equation for the El- lipse, subtracting 1 from each side forthe Hyperbola:

2c 2a

FIQ = pi '

That is FIQ ge epi .

Now if H be the foot of the perpendicularfrom Q upon a focal radius, Fill I= epicos

and PH = pi - epi.cos 9 = A = Ncos a. For the Parabola, the angles atP and Q are each equal to a and FA = pi. Thus PH = pi - woos 0 = A = Ncos a.

twcordingly, The projection of the NormalLength upon a focal radius is constant and equal tothe semi-latus rectum.

20. CENTER OF CURVATURE: Since ccc a =-4 from (19), N '

and IRI 11;, from(18),

we have IRI=Nsec2 a. WOOS 55 Thus to locate the center of curvature, C, draw the perpendicular to the normal at Q meeting a focal radius at K. Draw tae perpendicular at K to this focal radius meeting the normal in C. (For the Evolutes of the Conics, see Evolutes, 4.) (In connectionsee Keill.)

Fig. 55

BIBLIOGRAPHY

Baker, W. M.: Algebraic Geometry, Bell and Sons (1906) 313. Brink, R. W.: A First Year of College Mathematics, Appleton Century (1937). Candy, A. L.: Analytic Geometry, D. C. Heath (1900) 155. Graham, John and Cooley: Analytic Geometry, Prentice- Hall (1936) 207. Niewenglowski, B.: Cours de G4om4trie ApiejlyliguE, Paris (1895). Salmon, 0.1 Conic $ectons, Longmans, Green (1900). Sanger, R. G.: Synthetic Projective Geometry, McGraw Hill (1939) 66. Winger, R. M.: Projective Geometry, D. C. Heath (1923) 112. Yates, R. C.: Geometrical Tools, Educational Publishers, St. Louis (1949) 174, 180, Kern, J.: Philosophical Transactions, XXVI (1708-9) 177 -8. CUBIC PARALJOLA

HISTORY: Studied particularly by Newton andLeibnitz (1675) who sought a curve whose subnormal is inversely proportional to its ordinate. Monge used theParabola y = x3 in 1815 to solve everycubic of the form 3 x hx + k = O.

1. DESCRIPTION: The curve is defined by theequation:

y = Ax3 +Bx2 + Cx + D = A(x - a) (x2 + bx + c). 14,

b'- 4c >0 e_ 4c. 0

Fig. 56

2. GENERAL ITEMS: (a) The Cubic Parabola has max-min. points onlyif B2 - 3AC > 0.

(b) Its flex point is at x =:13, (a translation of 3m the y-axis by this amount removes the squareterm and thus selects the mean of the roots asthe origin). (c) The curve is symmetrical with respect toits flex point (see b.). (d) It is a special case of the Pearls of Sluae.

(e) It is used extensively as a transition curvein railroad engineering. CV= MAMA 57 (C) It is continuous for all values of x, withno asymptotes, cusps, or double points. (g) The Evolute of ey= x3 is

3a2(x2 2va)2 128,2 2 9 4 3 3 a - 243 4 125 125 xy)(:7 a - "2y - y) 2: 0 4 (114 x40 (h) For 3a2y= x3, R = 1 20x (1) Graphical and MechanicalSolutions: 1. Replace x3 + hx + k= 0 by the system: = x3 y + hx +k = 0, the abscissas of whose intersections are roots of the given equation. Only one Cubic Parabola need be drawn for all eubics, but for each cubic there is a particular line.

Fig. 57

2. Reduce the given cubic xI3 + hxi k = 0 by means of the rational transformation xi= x to the form ha x3 + m(x + 1)= 0 in which ma

* The diecriminant (the squaLe of the productof the differences of the roots taken in pairs) of this cubic is:

A = 412( 27 + 4m). Thus the roots are real and unequal ifm < . T7 j two are complex 27 if m j and two or more are equal if m 27 4 ft 0 or m = 17 Thee° regions of the plane (orranges or m) are separated by the line through (.1,0) tangent to tne curveas shown. CUBIC PARABOLA This may be replaced by the system ty=x3, y+m(x+1)=0). Since the solution of each cubic here requires only the determination of a x particular slope, a straightedge may be attached to the point (-1,0) with the y-axis accom- modating the quantity m.

Fig. 58

(j) Trisection of the Angle: Given the angle AOB ge 30. If OA be the radiusof the unit circle, then the projection a is cos30. It is proposed to find cos U and thus 0 itself. Since cos 30 m 4 0°830- 3 cos 0, we have, in setting x = cos 0: 4x3- 3x-aft0 or the equivalent systems y= 4x3, y 3x - a 0. Thus, for trisection of

0 30, draw the line through (0,a) parallel to the fixed line L of slope 3. This meets the curve y 4x3 at P. The line from P perpendicular to Pig, 59 CUBIC PARABOLA 59 OB meets the unit circle in T and determines the required distance x. The trisecting line is OT.

BIBLIOGRAPHY

Yates, R. C.: Geometrical Tools, Educational Publishers, St. Louis (1949). Yates, R. C.: The Trisection Problem, West Point, N. Y. (1942). CURVATURE

1. DEFINITION: Curvature is a measure of the pate of change of the angle of inclination of the tangent with respect to the arc length, Precisely,

K = A2 I

do '

AL a maximum or minimum joint K y" (or. 02,0)1 at a flex if y" is continuous, K = 0 (or 02); at a puap, R = O. (See Evolutes).

2. OSCULATING CIRCLE: The osculating circle of /'"-- a curve is the circle having (x,y), y' and y" in common with the curve. That is, the 77 ;I)

'1,..., 1 relations: (x a)2 (y 02 r2 \\\ oca) ., (x - a) + (Y - 0)Y' m 0

(1 4. Y12)4. (Y -0)Y" 00 must subsist for values of x,y,y',y" belonging to the curve. These conditions give: Fig. 60

r = R, ard X- R,sin 4,, 0 = y Ricos 11), where y is the tangential angle. This is also called the Circle of Curvature.

3. CURVATURE AT T1 ORIGIN (Newton): We consider only rational algebraic curves having the x-axis as a tangent at the origin. Let A be the centerof a circle tangent to the curve at 0 and intersecting the curve again at Pt(x,v). As P approaches 0, the circle approaches the osculating circle, Now BPm x is imean CURVATURE 61 proportional between OB = y and BC = 2R -y, where AO = R. That is,

x 2R - y = , and x2 Ro m Limit R = Limit (-4-). 2y P 0 x 0 y-.0

Examples: The Parabola 2y = x2 has Ro = 1. x2 The Cubic y2 = x3 or -- %has Ro= O. 2y 2 2 1 The Quintic y = X.or = ,.- has Ro = 6". 2y 2vx Generally, curvature at the origin is independent of all coefficients except those of y and x2. If the curve be given in polar coordinates, through the pole and tangent to the polar axis, there is in like fashion (see Fig. 61):

2R'sin 0 = r or R = 2 sin U

Ro = Limit r Limit (1 e - 0 '2 sin 0' 0 0'20)

&Eau: The Circle a(sin = a' sin 0 orr has Ro = 20 20 2 The Cardioid

b) r = 1 - cos 0 or -11- =(1 cos has Ro = O. 20 20 E.

62 CURVATURE 4. CURVATURE IN VARIOUS COORDINATESYSTEMS:

(1 z.2)3 R = de/dY . Y 2 R =r(2) dX 2 la 2 dp +.( ) ds da2 x2 R = p + .

Bo = Limit (--) . 2y x 0 (r2 + r'2)3 (polar 0 (r2 + 2r,2 - rr")2 coords.)

(i2 + Y2)3 )3 R2 (xy - 0'3062 2fxyfxfy + fyyfx2T2'

[where the curve is [where the curvo is f(x,y) = 0].

x = x(t), y y(t) and 3 N , where B2 (-1.1. pey" dt N2 y2(1 vs) 2 v R = , where v, aare an n (See Conics, 18). magnitudes of velocity and normal acceleration of a moving point.

5. CURVATURE AT A SINGULAR POINT:At a singular point of a curve f(x,y) = 0, fx =fy = O. The character of the point is disclosed by the form: F a fzy2 - fxxfyy. That is, if F < 0 there is anisolated Loint, if F = 0, a cusp, if r > 0, a node. Thecurvature at such a point (exclAing the case F < 0) is determined by theusual

11 after y' and y" have been evaluated. (1 + y1 ) The slopes y' may be determined(except when y' does not exist) from the indeterminate form-711 by the appropriate process involvingdifferentiation. CURVATURE 63 6. CURVATURE FOR VARIOUS CURVES:

IHVE:1 E:JJATION 11 o Rect., , 2 r r sin 2U . 2k Hyperbola 2k2 . 2 :' 2 , 4z._ 2 (See con- s'' Catenary r.: c .4. = c'secy c struction E. under Cate- nary)

Cycloid s 4 Oay (See construe- Ital1 - -1- 2a tion under Cycloid)

x = a(t - sin t) t y = a(1 -.10dt) .4e.ein (--2 )

Tractrix a = cln sec y c tan y

Equiangular s = a(e1114- 1) maellf Spiral

(See construction under Lemniscate r3=alp 3r Lemniscats) 2 22 a2 b2 r2 2___ ab Ellipse - 1- P2 P Sinusoidai an rn = a n cote nU Spirals (n + 1)rn'1 ((n + 1)p

a g Aatrold 3 3 x+ y . 3( ax Ali 3

Epi- and 2 = a sin by a(1-b)sin by. (142)p Hypo-cycloids

7. UNERAL ITEMS: (a) Osoulatilu; circles at two corresponding points of inverse curves are inverse to each other. (b) If R and R' he radii of curvature of a curve and its pedal at corresponding points:

R1(2r2 - pR) r3. 64 CURVATURE (c) The curve y = xn is useful indiscussing curvature. Consider at theorigin the cases for n rational, when n < > 2. (See Evolutes.) (d) For a parabola, R is twice thelength of the normal intercepted by the curveand its directrix.

BIBLIOGRAPHY

Edwards, J.: Calculus,Macmillan (1892) 252. Salmon, G.: Higher PlaneCurves, Dublin (1879) 84. CYCLOID

HISTORY: Apparently firstconceived by Mersenne and Galileo Galilei in 1599 andstudied by Hoberval, Des- cartes, Pascal, Wallis, theBernoullis and others. It enters naturally into a variety of situations andis justly celebrated. (See 4band 4f.)

1. DESCRIPTION: The Cycloidis the path of a point ofa circle rolling upon a fixedline (a roulette). The Prolate and Curtate Cycloidsare formed if P is not on the circle but rigidlyattached to it. For a point-wise

Pig. 62 construction, divide the interval OH (= na) andthe semicircle NH into an equal number of parts: 1,2, 3, etc. Lay off 1P1= H1, 2P2 = H2, etc., as shown.

x a(t- sin t) y = a(1 - cos t)=2a.sin2*. s = 4a.sin 0 R2 + s2= 16a2. (measured from top of arch). 66 CYCLOID 3. METRICAL PROPERTIES:

(a)y ("2 2 (b) L(one arch) = 8a (since Ro = 0, RN = 4a) (SirChris-.

topher Wren, 1658).

(c) y' = cot* (since H is instantaneous center of rotation of P. Thus the tangent at P passesthrough N) (Descartes). (d) R = 4a.cos 0 = 4asin() = 2 (PH) =2(Normal).

(e) s = 4a.cos) = 2(NP).

(f) A(onearclo = 3naa (Roberval 1634, Galileoapproxi- mated this result in 1599 by carefully weighing pieces of paper cut into the shapes of acycloidal arch and the generatingcircle).

4. GENERAL ITEMS: (a) Its evolute is an equal Cycloid. (Huygens 1673.) (Since s = 4a.sin 0, a = 4a.cos0 = 4a.sinP) R = PP' (the reflected circle rolls along the hori- zontal through 0'. P' describes the evolute cycloid. One curve is thus an involute (or the evolute) of the other. (See Evolutes, 7.)

rig. 63 CYCLOID 67

t ds t, (b) Since s 4a.cos(-2 ), = lay . dt (c) A Tautochrone: The problem of the Tautochrone Is the determination of the type of curve along which a particle moves, subject to a specified force, to ar- rive at a given point in the same time interval no matter from what init!al point it starts. The following was first demonstrated by Huygens in 1673, then by Newton in 1687, and later discussed by Jean Bernoulli, Euler, and Lagrange. A particle P is confihed in a vertical plane to a curve s = f(y) under the influenceof gravity: ms = -mwsin T.

Fig. 64

If the particle is to produceharmonic motion: ms = -les, then

s = (')sin T,

that is, the curve of restraint mustbe a cycloid, generated by a circle of radius11/1k. The period of this motion is 2n, a period whichis independent of the amplitude. Thus two balls(particles) of the same mass, falling on acycloidal arc from different heights, will reach the lowest pointat the same instant. 68 CYCLOID Since the evolute (or an involute)of a cycloid is an equal cycloid, a hob B may be supported at 0 to de- 0 scribe cycloidal motion. The period \\\ of vibration of the

. ,\\ pendulum (under no .' . resistance) would be constant for all amplitudes and thus Fig. 65 the swings would cuur.t equal time intervals.Clocks designed upon this prihciple were short lived.

(d) A Brachistochrone. First proposedby Jean Bernoulli in 1696, the problem of theBrachistochrone is the determinatic,of the path along which a particle moves from one p.cint in a planeto another, subject to a specified force, in the shortest time. The following discussion

tt Is essentially the solution given by h Jacques Bernoulli. Solutions were also Fig. 66 presented by Leibritz, Nr!wtLA), aLd 3111.)cp1 tal.

For a body rallirr* under gravity alongany curve of rest-Rint: y = g, y = y = or t =1137 At any instant, the velocity of fallis CYCLOID 69

=8.17Ei Vr25-.

Let the medium through which the particlefalls have uniform density. At any depth y, v = V2gy . Let theoretical layers of the medipm be of infinitesimal depth and assume that the velocity of theparticle changeS at the surface of each layer. If it is to pass from Po to PI to P2 in shortest time, then according to the law of refraction: sin al stn ag sin a3 ..4 vITT /747i1 V 6gh Thus the curve of descent, the limit ofthe polygon as h approaches zero and thr numberof layers increases accordingly), is such that (Fig. 67): sin a = PVT or cos2U = k2y, an equation that may beiden- tified as that of a Lycloid. (e) The parallel projection of a cylindrical helix onto a plane perpendicular to its axis is a Cycloid, prolate, curtate, or ordinary.(Mon- rig.67 tucla, 1799; Guillery,1847.) (f) The Catacaustic of a cycloidalarch for a set of parallel rays perpendicular toits base is composed of two Cycloidal arches.(Jean Bernoulli 1692.) (g) The isoptic curve of aCycloid is a Curtate or Prolate Cycloid (de La Hire17C4). (h) Its radial curve is a Circle. (I) It is frequently found desirable todesign the face and flank or teeth Inrack gears as Cycloids. (Fig. 68). 70 CYCLOID

Pig. 68

BIBLIOGRAPHY

Edwards, J.: Calculus, Macmillan (1892) 337. Encyclopaedia Britannica: 14th Ed. under "Curves, Special". Gunther, S.: Bibl. Math. (2) vl, p.8. Keown and Faires, Mechanism, McGraw Hill (1931) 139. Salmon, G.: Higher Plane Curves, Dublin (1879) 275. Webnter, A.G.: Dynamics of a Particle, Leipslg (1912) 77. WA.ff[ng, E.: Binl. Math. (3) v2,p.235. DELTOID

HISTORY: Conceived by Euler in 1745 inconnection with a study of caustic curves.

1. DESCRIPTION: The Deltoid is.a 3-cuspedHypocycloid, The rolling circle may be either one-third(a = 3b) or two-thirds (2a = 3b) as large as the fixed circle.

Fig. 69

For the diouble generation, considerthe right-hand Pa figure. here OE = OT = a, AD = AT = ":,where 0 is the 3 .enter olf the fixed circle and A thatof the rolling circle which carries the tracingpoint P. Draw PP to T', T'E, PD and T10 meeting in F. Drawthe c!rcumcircle of F, P, and T' with center atA'. This circle is tangent

-;-1 and its t,1 the fixed circle at T' sinceangle FPT' = diameter FT' extended passes through O. Triangles TET', TDF, and T'FI are allsimilar and 72 DELTOID Tp = . Thus the radius of this smallest circle is T'P 1 Furthermore, arc TP arc T'P = arc TT'. itcordingly, if P were to start at X, either circle would generatethe same Deltoid -the circles rolling in opposite direction. (Notice that PD is the tangent at P.)

2. EQUATIONS: (where a = 30. {x=b( 2 cos t + cos 2t) ( x2+y2)2+8bx3 - 24bxy2 + 18C( x2+y2)ft 27b4. y = b( 2 sin t - sin 2t).

s =(112)cos )y. R2 + 9s2= 64b2. r2 3 = 9b2 - 8p2. p = h.sin 34. E =b(Loit 0.'21+).

3. METRICAL PROPERTIES:

t ds L lob. R -8p. = dy A. 2nb2 = double that of the inscribed ylriv, 4b = length of tangent (BC) inter,:eptedhy

4. GENERA?. ITEMS:

(a) It is the envelope of the SLMSOtt uf a fixed triangle (the line formed by the feet of theperpen- diculars dropped onto the sides from a variable point on the circumclrcle). The center of the curve is at the center of the triangle's nine-point-circle. (b) Its evolute is another Deltoid. (c) Kakeya (1) conjectured that it encloses a region of least area within which a straight rod, taking all possible orientations in its motion, can be reversed. R3wever, Beslcnvitch showed that there is no leant area (2). (d) Its inverse is a Cotes' Spiral. (e) Its ocial with respect to (c,0) is the family of folia

f(x - c)2 + y2jfe + (x c)x) = 41)(x - c)y2 DELTOID 73 (reducible to: r = In) cos usIn`u c.cos U) (with respect to a cusp, vertex, or center: a simple, double, trl-follum, rasp.). (f) 291119nt Con)Aruction: Since T is the instantaneous center of rotation or P,`PP is normal to the path. The tangent thus passes through N, the extremity of the diameter through T. (g) The tangent lenPth intercepted by the curve is constant. (h) The tangent BC is bisected (at N) by the inscribed

(I) 1:r catacaust12 for a set or parallel rays is an Astrold. (j) Its urthoptic curvg.,is a Circle. (the inscribed circle) . (k) Its radial curve Is a trifolium. (1) It is the envelope of the tangent fixed at the vertex or a )arabola which touches riven lines (a Roulette). It is also the envelope of this Parabola. (m) The tangents at the extremities B, C meet atright angles on the inscribed circle. (n) The nor:Ials to the curve at B, C, and P all meet at T, a point of the cIrcumcIrcle. (o) If the tangent BC be held fixed (as atangent) and the Deltoid allowed to mwe0 thelocus of the cusps is a Nephroid.(For an elementary geometrical proof of this elegant property, see Nat.Math. Mag., XIX (1945) p. 330. 74 DELTOID BIBLIOGRAPHY

American Mathematical Monthly, v29, (10,,2) 160; v46, (1939) 85. Bull. A. M. S., v28 (1922) 45. Cremona, Crelle (1865). Ferrers, kw. Jour. Math. (1866). L'Interm. d. Math.,v3, p.166; v4, 7. Morley and Morley: Inversive Geometry, G. Bell and Sons (1933). National Mathematics Magazine, XIX (1944-5) 327. Proc. Edin. Math. Soc., v23, 80. Serret; Nouv. Ann. (1870). Townsend, Educ. Times Reprint (1866). Wieleitner, H.: Spezielle ebene Kurven, Leipsig (1908) 142 (1) Tohoku Sc. Reports (1917) 71. (2) Mathematische Zeitschrilt (1928) 312. ENVELOPES

HISTORY:Leibnitz(1694) and Taylor (1715) were the first to encounter singular solutions of differential equations. Their geometrical significance was first indicated by Lagrange in 1774. Particular studies were made by Cayley in 1872 and Hill in 1888 and 1918.

1. DEFINITION: A differential equation of the nth degree

,* X defines n p's (real or imaginary) 4 )1 Cor every point (x,y) in the plane. Its solution F(x,y,c) = 0, r of the nth degree in c, defines X n c's for each (x,y). Thus attached to each point in the plane there are n integral curves with n corresponding slopes.Throughout Fig, 70 the plane some of these curves together with their slopes may be real, someimaginary, some coincident. The locus of thosepoints where there are two or more equalvalues of p, or, which is the same thing, two or more equal values of c, is theenvelope of the family of its integral curves. In otherwords, this envelope is a curve which touches at eachof its points a curve of the family. Theequation of the envelope satisfies the differential equation but isusually not a member of the family.t

p is used here for the derivativeto conform with the general cuatom throughout the literature. Itshould no: be confused with the dint/ince from origin to tangent anused eleWhere in thin book. tlhe line y 0 to a pert or the envelope and a member of the family y c(c-4x)2 76 ENVELOPES Since a dcuble root of an equation must also be a root of Its dcrIvatIve (and conversely), th envelope is obtained from either of the sets (thf dli:rant relation):*

f(x,y,p)=0 F(x,y,c)= 0

[fp(x,y,p)=0 Fc(x,y,c)= 0 Each of these sets constitutes the parametric equations of the envelope.

2. EXAMPLES: 4 :7 y-px - -= 0 P

4 x + M =.0. VP ' P

F '-- y -cx - -1-- = 0

Fc 7 7+..M _0. C lye yieldinv: t= 16x as the envelope,

Fig. 71

f . - px - 2--- 0 Y (p-1 )

fp - (17).-.11- = 0.

F x'soc2u y'csc2u- 1= 0 Fc 2xasec2u tan U

2yacsc2u cot u=0 .

yields r:,' the parabola + = Fig. 72 as the envelope of lines, the sum of whose Intercepts is a positive constant.

Such questions na tats locuu, ,,uapilal rind nodal loci, !tc., whose eontionu nppenr aa factors in one or both discriminanta, are die- Cusa$1 in Rill (191A). For exarplun, nee Cohen, Murray, Olaisher. ENVELOPES 77 MUTE: The two preceedir: examples are differential equations of the Clairaut form: = PY +

The m,.t.h.d s,lution is that of differentiating w!th res.pe.t to x:

P + x(-1dx 2) (dp -11:)(-(-12) P dx'

Hence, (-1:1)1qx + (IC)]= 0, and the general solu- dx dp

tion !r1 oobtained from the first factor:112= 0, or dx p = c. That is, y = CY g(c). dE seccnd Castor: x = 0 is recognized as dp fp = 0, a requirement for an envelope.

3. TECHNrQUE: A family of curves may begiven in terms or two parameters, a, b, which, themselves, are connected by a certain relation. The following method is proper and is part:cniarly adaptable to forms which are homogeneous in tit parameters. Thus

give.; :(x,v,a,0 0 and e(a,b) = 0. Their part:al d!l'ferentials are

facia + Vbdil = 0 and gada + gydb = 0 and thus fa = Ngn, fb = where is a factor of propot:onality to be determined. The quantities a, b may be eliminated amoni!, the equations to give the envelope. Per example: (a) Consider the envelope of a line of constant length mcvin ! wIth its ends upon the coordinate axes (a Trammel of Archi-

medes): + Y= 1 where a b a2 + b2 = 1. Their differentials

give (fl ,)da + ("is)dn = 0 rn.ci b- a.da + b.db = 0.

b, a'- Fig. '(3 78 ENVELOPES Multiplyln: tht. first by a, the second by b, and add- x i =1 x(a2 I-j21 = x, by virtue or the, a given functons. Thus, - 1 and a` + b2 = 1,

x = a3, y = b3, or xl+y5=-71]an Astroid,

(b) Consider concentric and coaxial ellipses ofcon- s 2 V stant area:-T+ 4-T = 1, where a y2 1/2 ab = k. We hhve (-s)da + V-5)db =0, a bda + ndb = 0, from whicn 2 2 = Xb, xa. Multiplying the

first by a, t.'u second by b: and addirw: 1 = ak and thus X 2k. , a pair of Fig. 71+

A w.dHh. WL, paper. C H o: r and P a r. .t Fold P ti: circle P' at:,1 P' upon. L11:. 'Ircle, the creases onv-.1ope a ,p:.tral P and C as roc::

Fig. if

ENVELOPES 79 an El]lpse if P be inside the circle, anHyperbola if outside. (Draw CP' cutting the crease in Q. Theh PQ = P'Q u, QC = v. For the Ellipse, u + v = P;for the Hyperbola u -v = r. The creaser; are tangentssinge they bisect the angles formed by the focalradii.) For the ParaboTh, a fixed point P ts rohied over to P' upon a fixed line L(a circle of infiniteradius). P'Q is drawn perpendicular to L and, since PQ =P'Q, the locus of Q Is the Parabola with P as focus,L as directrix, and the crease as a tangent.(The simplicity of this demonstration should be compared to ananalytical method.) (See Conics 16.)

5. GENERAL ITEMS: (a) The Evolute of a _liven curve Is the envelope of Its normals. (b) Tne Catacaustic of a given curve is the envelope of it3 reflected light rays; the Diacausticis the envelope of refracted rays. (c) Curves parallel to a given curve may be considered as: the envelope of circles or fixed radius with centers on t!o given curve; or as the imvelope of circles of fixed radiustangent to the given curve; or as the envelope of lines parallel to thetangent to the given curve and at a constantdistance from the tangent. (d) The first positive Pedal of a given curve isthe envelope or circles through the pedalpoint w!th the radius vector from the pedal point asdiameter. (e) The first negative Pedal is the envelope of the line through a point of the curveperpendicular to the radius vector from the pedalpoint.

(f)if L, M, N are linear functionsof x,y, the envelope of the family Lc4 + 214i.c +N = 0 in the conic

Fte= L'N '1 80 ENVELOPES where L = 0, N = 0 are two of its tangents and M = 0 their hord of contact. (Fig. 76). (g) The envelope of a line (or curve) carried by A curve rolling upon a fixed curve is a Hcrolette. For example: the envelope of a diameter of a circle rolling upon a line is a Cycloid;

Fig. 76 the envelope of the directrix of a Parabola rolling rpon a line is a Cate- nary: (h) An important envelope arises in the following calculus of variations problem (Fig. 77): Given the eurva r = 0, the point A, both in a plane, and a constant Y2 C force. Let y = c be the line F.0 of zero velocity. The shortest tire path from A to F 0 is E0 A the Cycloid normal to F = 0 generated by a circle rolling upcn y = c. However, let the family of Cycloids normal to F = 0 generated by all circles rolling upon y = c envelope the curve. E = O. f this envelope Fig. 77 passes between A and F = 0, there is no unique solution of the problem.

BIBLIOGRAPHY

Bliss, G. A.: Calculus of Variations, Open Court (1935). Cayley, A.: Mesa, Math., II (18721. Clairaut: Mem. Paris Acad. Sci., (1734). CoherA.: Differential Equations, D. C. Heath (1933) 86 -100. Glaisher, J. W. L.: Mess. Math., XII (1882) 1-14(examlaes) Hill, M. J. M.: Proc. Lond. Math. Sc. XIX (1888) 561- 589, ibid., S 2, XVII (1918) 149. Kells, L. M. t Differential Equations, McGraw Hill (1935) 75rf. Lagrange: Mem. Berlin Acad. Sci., (1774). Murray, D. A.: Differential Equations, 7',ongmans, Green (1935) 40.49, EPI- and HYPO-CYCLOIDS

HISTORY: Cycloidal curves were first conceived by Roemer la Dane) in lo74 while studying the best form for gear teeth. Onlileo and Mersenne had already (1599) discovered the ordinary Cycloid. The beautiful double genera- tiori theorem of there curves was first noticed by Daniel Bernoulli in 1725. Astronomers find forms of the cycloidal curvet in various coronas (see Proctor). They also occur as Caustics. Rectification was 4 .,iven by Newton in hit Principia.

1. DESCRIPTION: The Epicyclo'd is gen- The Hypocycloid is generated r: a point of a erated by a point of a circle ro.lin extornall2 circle rolling internally upon a fixed circle. upon a fixed circle.

Pig. 78

2. DOUBLE GENERATION: Let the fixed circle have center 0 and radius OT = OE = a) and the rolling circle center A' and radius 82 EPI- and HYPO-CYCLOIDS A'1" = A'P = 6, the latter carrying the tracing'point P. (Set Pb. r79.) Draw ET', oTT, anu PT' to T. Let D be the interse.'t: of TO and FP and draw the circle on T, P, and D. Th:s circle is tanr,ent to':he Ci>ed circle since angle DPT is a rlitt afwle. Now sin-:e PD Is parallel to T'E, triangles LET' and oFD are isosceles and thus DE . 2b. Furthermore, arc TT' = au and are T'P = bu arc T'X. Accordtr-ly, arc. TX = (a b) u = arc TP, ror the Eyipyclold,

Or' = (a - b) U . arc PP, for the Hypocycloid. Thus, each of these sycloidal curves may be generated in two ways: by two roll121g circles the sum, or difference, of whose radii is the radius or the fixed circle.

Fig. 79

The theorem is also evident from the analyticviewpoint. Consider the case or the Hypocycleid:(Euler, 1784)

t . (a -')cos t b'cos(a

y = (a b) sin t h.sin(a -4.1 r, 77-11317717'7,T,,,,77,7-7'r7,15,77-.

EP!- and HYPOCYCLOIDS 83

(a + c) (a+ c.)ti andletb = ---7----,t Thy, 2 = ---- . equations brwome: (drot.....iv, JuorliA) r JE1.-,:!)1,s (a+ ,:.)t (a+4con (a-c)t---- (!

r117_.211sLn 1.12.211L (a:c) . (a-,..,)t x.' = ["1 2 ' 0 2 Noti,:cthat a Lllani.!cin aim or c does not alter these equations. Accordincly, rollin..: circles of radii(a + .....--c) or(a c)b,,enerate the same curve upon a fixed circle of radius a. That is, the difference of the radii of fixed circle and rollini,7 circle iri.ves the _radius of a third circle which will mneratf: the same imilcloi.d. An analc,ous demonstration for the Epicz;1.old can be constructed w.i thou t,cri rficulty. 3. EQUATION:3:

EPTCYCLOID HYPOCYCLOID t xv (a +b)coa t-bLos(a+1) 1 x .. (a-b)cos t +bcos(a-b) b

y .,... (a+b)olu t bsin(a+b)- y ...(a-b)sin t - b'sin(a-b)7.

(X-axlc thrugh (x-axis through a cusp)

x. (a+b)..:os t + b'cos(a+o)- = (u-b)coa I -hcos(a-b)! U b

I; z (u+b)utn t + b'oin(a+b)- y= (a -b)uinI + b'Sin(a-b)- , b (x -axis bisectingarc betwetn 2 successive cusps)

Lib(a k1) a 4.Lb_.:L.L.q. a.-...:-...... -tiLii a. 8 --.-...-. (F a t, sin a a + 2b u a - 2b-- q., OP

[ Aain F3'1

where 13 < 1 Epicyclold,

13 = 1 Ordinary Cycloid,

13> 1 Hypocycloid. *.ft';iis equation, A' ,:ourse, maw (lust as well iLvolve the cosine. 84 EPI- and HYPO-CYCLOIDS

[R2+ 8262 = A2B2

2 (32(1.2 a2) r2= a2+ 4/1122.n1 km + 1)- [I 2b)2 wheree 4b(a + b) (a - 2b)2 or = 4b(b - a) '

(a + h) where m for the Epicycloid

(b - a) M = for the Hypocycloid.

1Bp= a'sin By

4. METRICAL PROPERTIES:

% 8b2k a + b) L (of one arch) = where = or a A (of' segment formed by one arc:. and thecenter) 2 na .1 = k(k where k has the values above. (k1) 4p R = AB'cos By = with the foregoing values of (k +1)g k. (y may be ohtaintql in terms of t from the given figures) . (See Am. Math. Monthly (1944) p. 587 for anelementary demonstration of these properties.)

5. SPECIAL CASES:

Epicycloidst If b = a...Cardioid 2b = a...Nephroid.

Hypocycloids: If 2b = a...Line Segment (SeeTrochoids) 3b . a...Deltoid 4b = a...Astroid. EPI- and HYPO-CYCLOIDS 85 6. GENERAL ITEMS: (a) The Evolute of any Cycloicial Curve is another of the same species. (For, since all such curves are of ds the form: s = A sin By, their °volutes are = a = dy AB in By. These evoluces are thus Cycloidal Curves similar to their involutes with linear dimensions altered by the factor B. Evolutes of Epicycloids are smaller, those of Hypocycloids larger, than the curves themselves). (b) The envelope or the family of lines: x cos u y sin U = c.sin(nU) (with parameter U) is an Epi- or' Hypocycloid. (c) Pedals with respect to the center are the Rose Curves: r = c.sin(nU). (See Trochoids). The lsoptic of an Epicycloid is an Epitrochoid (Charles 183y). (e) The Epicycloids are Tautocnrones (see Ohrtmann). (f) Tangent Construction: Since T (see figures) is the instantaneous center of rotation of P, TP is normal to the path of P. The perpendicular to TP is thus the tangent at P. The tangent is accordingly the chord of the rolling circle passing through N, the point diametrically opposite T, the point of contact of the circles.

BIBLIOGRAPHY

Edwards, J.: Calculus, Macmillan (1892) 337. Encyclopaedia Britannica, 14th Ed., "Curves, Special". Ohrtmann, C.: Das Problem der Tautochronen. Proctor, R. A.: The Geometry of Cyclolds (187e). Salmon, O.: yiglv Plane Curves, Dublin (1879) 278. Wieleitner, H.: Spezielle ebene Kurven, Leipsig (1908). itTrr.7,E1771r"F

EVOLUTES

H1STCRY: Idea of evolutes reliutedly uii,!nated with EFly;,,ens in ,on:,en with his studies on light. H,.wevor, tile ma:( be tr A to Apollonius (about

_q.10 BC) where :t appears .1 ILL fifth book (if his Conic Suot ;ors,

1. DEFINITION: The Evulute of a curve is the locus of Its centers of curvature. If (a,p) is this center, a = x- R.sin T,

p y + Rcos p, where R is the radius of curvature, y the tangential angle, and (x,y) a point of the given curve. The quantities x,y,R,sin y, cos y may be expressed in terms of a single variable which acts as a parameter in the equations (in a,p) of the evolute.

Fig,. 80

2. IMPORTANT RELATIONS: It' s isthe arc length of the given curve,

da dx R cos y(dy/ds) -sin 4(-1y;), ds ds

R sin y(dy/ds) + cospg). ds ds ds dx ds But sin y =ds cos R dp .

da dR, Thus -sin y(--), . cos4.11. ds ds ds ds'

1.1.0 _ = -cot y . Hence da EVOLUTES 87 Lanoents to the evAute art nnrmals to the cu, fn ,:ther words, the evolute Is the enveluif L,raji:. *.c r!ven

Prom the

d . 4 dR where da2 = (ice + de,

Thus I cr = R i

That IL ;,Lhe of the evolute (Jr R is mknotone) is the dLrference of ti,p radt: LZ curvature of the fr:ver curve measured yrom the end IntF. el: the are a Furthermore, :ver, curve is an rIvuluto

Fig. 81

3. GENERAL ITEMS: rMany or these may be estal)lished most s!roply by ustnr the Whewell equation, of the curve. See 3e.. 7 rC.] a :em1:Ihtc

(b) The evu.:ute of a central coniL: the Law; curve:

(A) +(1) = 1. - B (c) The evolute or an equiangular spiral is an equal equiangular (d) The evolute of a Tractrix is a CatenarI. (e) Evolutes of the Epi- and Rypocycloids are curves of the same specie3. (See Intrinsic Eqns. and 4(b) following.] (f) The evolute or a Cayiel !!er.te is a Nephroid. (1') The Cata,.aust'e of a phien curve evolute of ort.ftJt_ (Sec Caustic:3.) (h) Gnevall::, to a flex point or: a curve corresponds an asymptote toit, evolute. (For exceWon see y3 = xs, 4(c) roll:dwing.] 88 EVOLUTES 4. EVOLUTES OF SOME CURVES: (a) The Conics:

The Evolute or 2 d i i

The Ellipse: () + (az) ..:, 1 is (21) 4.(2) . 1 a 1. A B Aa = Bb = a2 - be.

2 2 i

The Hyperbola: (!) -(i) = 1 is(i) - (K) = 1 ,

Ha = Kb = a2 + b2.

The Parabola: x2 = 2ky is x2 m 47; (y - k)3 .

(An elegant construction for the center or Curvature of a conic i' given in Conics 20.) E VOL UTES 89 (h) The Cycloids (their evolutes are of the same species):

Rocs 1' s: cosn .

sact

tY. 41 cost

sfi

Br; r.os 5' X ty" ;1sIn 2t 4

Fig. 83

(c) The Family y = xn. If the x-axis is tangent at the origin: .2 2-n Ro = Limit (-'2) Limit (=--). [See CurvatureCurvature.] 2y 2 Thus: Ro = 0 if n < 2; Ro = co if n > 2; 1 Ro = if n = 2. 2 E VOL UTES

P163.

5. GENERAL NOTE: Where there Is symmetryIn the given :111-ve with respect to a line(except for points of osculation or double flex) there will correspond a cusp in the evolute (approaching the point ofsymmetry on either sld-, the normal forms a double tangentto the evolt,Le). This is not sufficient, however. if' a curve has a cusp of the first kind,its evolute in genc.ral passes through the cusp. If a oore has a cusp the second kind, there corresponds a flex in the E.volute. E VOL UTES 91 6. NoRMALS TO A GIVEN CURVE: The Evolute of a curve separates the plane Into regions containing points from which normals ma:, be drawn to the curve. For example, consider the Parabola y2 . 2x and the point (h,k). The normals rPOM (h,k) are determined from

y3 + 2(1- h)y 2k = 0, where y represents the ordinates of the feet of the normals at the curve. There are thus, in general, three normals and at their reet:

Y1 Y2 Y3 = 0. If we ask that. two of the three normals be coincident, the fore;oinc* ;:ublo must have a double root. Thus between this ,!uh::: ank !ts derivative; 3y2 + 2(1 - h) = 0, are the cohditIons on h and k: 7 2 h -t 1 + 21'k -y3. 2 The locus of (h,k) Ls t:els recognizable as the Evolute of the given Parabola: the envelope of tts normals. This evolute divides the plane into two regions from which one or three normals may be drawn to the Parabola. From points on the evolute, two normals may be estLblished.

An elegant theorem is a consequence of the preceding. The circle x2 y2 + ax + by + c = 0 meets the Psrabola y' = x in points such that

Y1 + Y2 Y3 Y4 = 0. If three of these points are feet of concurrent normals to the Parabola, then y4 = 0 and the circl? must necessarily pass through the vertex. A theorem involving the Cardiold can be obtained here by inversion. 92 EVOLUTES 7. INTRINSIC EQUATION OF THE EVOLUTE: Let the given curve he n = with the points 0' and P' of Its evolute corresponding to 0 and P of the given curve. Then, if a is the are length of the evolute: ds a -R -R=f'(q) -Ro. p 0 0 In terms of the tangential angle 0, (since p = q +t):

Fig. 85 a t."(0 1i) Ro

[Example: The Cycloid: s 4a.sin (f; a = Ita.cos y 4tcos(0 - 2) = 4a.oin pl. (See Cycloid 4,a).

BIBLIOGRAPHY

Byerly, W. E.: Differential Calculus,Ginn and Co. (1879). Encyclopaedia Britannica, 14th Ed. under"Curves, Speclal." &wards, J.: Calculus, gacmillan(1892) 268 ff. Salmon, G.: itahfr Plane Curves, Dublin(1879) 82 ff. Wieleitner, H.: Spezielle ebene Kurvon,Leipsig (1908) 169 ff.

6 EXPONENTIAL CURVES

HISTORY: The number "e" can be traced back to Napier and the year 1614 where It entered his system of logarithms. Strangely enough, Napier conceived his idea of logarithms before anything was known of exponents. The notion of a normally distributed variable originated with DeMoivre in 1733 who made known his ideas in a letter to an acquaintance. This was at a time when DeMoivre, banished to England from France, eked out a livelihood by supply- ing information on games of chance to gamblers. The Bernoulli approach through the binomial expansion was published posthumously in 1713.

1. DESCRIPTION: "e". Fundamental definitions of this important natural constant are:

e limi. , 1.1c limit x ("L +X) 0(1 x)I

57 Ta 2.718281 . 0

Fig. 86 94 EXPONENTIAL CURVES 2. GENERAL ITEMS: (a) ',ne dollar at 100.1' !nterest compounded k times a year rroduces at the end of tit:. year

kk-1) 1 k(k-1)(k-2) (1+1)k= Sk 2! 7" dollars. IC the interest be compounded continuously, thetotal at the end of the year Is

LL r. 1,i4,.11n,1 (1 +t)k $2.M.

(b) The Euler Corm: Ix e 16sin x 1 produces the numerical relations:

in 15 e 1 = 0 , e = 1. From the latter

7 i a (/=11 =(e1-2) = e 1- 0.208.

3. The Law of Growth (or Decay) is the productof exper- ience. In an ides' state (one in which there is no disease, pestileh 3, war, famine, or thelike) many natural population' increase, at a rate proportionalto the number present. Tnat Is,:C x repr-:.ents the ntaber

of individuals, and t 11.2 time,

dx = kx or dt cekt This occurs in controlled bacteria cultures,decomposi- tion and conversion oC chemical substances(such as radium and sugar), the accumulation of interestbearing money, certain types cq' electricalcircuits, and in the history of colonies such as fruit f11,.o andpeople.

A further hypotl:i :suppr:; the ioverning law as

dx = k.x.(n-x, x = (c e-nkt) t EXPONENTIAL CURVES 95 whore n Is the maximum possible number of inhabitants - a number regulated, for instance, by the food supply. A more general form devised to rtt observati9ns Involves the function r(t) (which may be periodic, for example): dx en = f(t)x.(n - x) or x . (Fig. 87a) f.dt (e + e

AL moderate velocities, the resistanceoffered by water to a ship (or air to an automobileor to a parachute) is Approximately proportional to the velocity. Thatis,

a = § = v = k2v, or s = (2?)(1 - 0-1tt).

s Z.,

(al Fig. (h)

4. THE PROBABILITY (OR NORMAL, OR GAUSSIAN) CURVE:

(Fig. 87b).

(a) Since y' =.-xy and y" = y(x2 -1), the points are (+1, e-1/2). ,Jo bed rectan w!th one side on the x-axis has area = xy = largest one is given by y" = 0 and thus two corners are titthe flex points.) 95 EXPONENTIAL CURVES (b) Area. By definition r (n) = zn-lezdz. In this, /20 co 2 211-'1 X2 let V (n) = x211-2.e 'I .2x dx = 2if X .e IdX. 0

Putting n = '

2 ir2 and I'() 2 dx = 47= Area between y = e

its asymptote.

The Normal Curve is, morespecifically: (x1)2 20 y a.71 e

For this population, n isthe size, g the mean, and the standard deviation.Rewriting for simplicity: 2/2a2 -x = kie

(+ a,Y0). It is evi- the flex points are(+ a, k'e-7) - dent that the flextangents:

a) - Yo (21g)(x- have x-intercepts which arecompletely independent or the selected y-unit. A stream of shot entering the "slot machine" shown is separated by nail ob- structions into bins. The collection will form into a histogram approximating the normal curve, the num. bee of shot in the bins proportional to the cuefficients in a binomial expansion.

rig.38 EXPONENTIALCURVES

BIBLIOGRAPHY

Kenney, J. F.: Mathematics of Statistics,Van Nostrand II (1941) 7 ff. Riet2, H. L.: Mathematical Statistics, OpenCou t (1926). Steinhaus, H.: Mathematical Snapshots,Ctechert (1938) 120. S.

FOLIUM OF DESCARTES

HISTORY: First discussed by DescartesIn 1638.

1. EQUAT1uNS: 3 3 X y yixy

:5at or J.) X (1 + t3) Q locp + -co lower- upper 3at2 (1 + )e.sln U cos r (;riTu c(43

Fig. 89

2. METRICAL PROPERTIES:

(a) Area ot' loop: ;A-2 area between curve and 2 asympt,te. FOLIUM OF DESCARTES 99 3. GENERAL: (a) its asymptote is x + y i a = 0.

(b) Its Hessian is another Polium of Descartes.

BIBLIOGRAPHY

Encyclopaedia Britannica, 14th Ed. under "Curves, Special." FUNCTIONS WITH DISCONTINUOUS PROPERTIES

This collection is composed of illustrations which may be useful at various times as counter examples to the more frequent functions having all the regular properties.

1. FUNCTIONS WITH REMOVABLE DISCONTINUITIES:

(a) y , undefined for y x = 2, is represented by the line y = x + 2 except for the point where x = 2. Since Limit y = 4, x 4 2

2 this is a removable discontinuity.

Fig. 90

12(2 -1 (b) y- , undefined for

x = 1, is represented by the Para- bola y = x2 + r. + 1 except for the point where x = 1. Since Limity =3, x4l this Is a removable discontinuity.

Fig. 91 FUNCTIONS WITH DISCONTINUOUS PROPERTIES 101 (c) The important function

sinX = , undefined for x = 0 has Limit y = 1 x 0 and thus has a removable discontinuity. The hyperbolas xy 1 form a Fig. 92 bound to the curve.

(d) The function y = xsin(-)1, is not defined for x = 0. However, Limit y = 0 x 0 and the function has a removable discontinuity at x = 0. The lines y = + x form a bound to the curve near x = 0.

Fig. 93 102 FUNCTIONS WITH DISCONTINUOUS PROPERTIES 2. FUNCTIONS WITH NON-REMOVABLE DISCuNTINUTTlES:

1 (aly = art! tar ; ,undefined

for x = 0. n Limit y Limit 2 x> 0+ x 0- x The left and right limits are both finIte but different.

Fig. 9h

(b) y = sin* is not

defined for x = 0. In every neighborhood of x = 0, y takes all values between +1 and -1. The x-axis is an asymptote.

1, Limitsin(X)does not x 0

exist.

Fig. 95 FUNCTIONS WITH DISCONTINUOUS PROPERTIES 105 (c) y = UmIt. (1 + sinny)t + I

( 1"!r ny )t -

Is discort:nlIbuc tY

+ X fl, 1, ;), 1 but ha o values il ur -1 c,lse- where.

Fig. 96

1 (d) y = 2x is undefined for

x = 0. Limit y = 0; x 0- Limit y m Left and right x 0+ limits different.

Fig. 97 104 FUNCTIONS WITH DISCONTINUOUS PROPERTIES

(0) Y- 2x + 1 is undefined for x = O. Since Limity = 1, and x 0- Limity = 0, left and x 0+ right limits at x = 0 a.e both finite but different.

Fig. 98

3. OTHER TYPES OF DISCONTINUITIES: (a) y = xx is undefined for

r. = 0, but Limity = 1. x i 0+ The function is everywhere dis-

X continuous for x

Pig. 99 FUNCTIONS WITH DISCONTINUOUS PROPERTIES 105

1 (b) y = xx is undefined for x = 0, but Limit y = 0. x 0+ The function is everywhere discontinuous for x < G.

" ......

......

Fig. 100 106 FUNCTIONS WITH DISCONTINUOUS PROPERTIES Cc) By halving the sides G., AC and CB of the .sos,!eles triangle ABC, and continuing this process as shown, the "saw tooth" path between <5,/ <>1 .A and B is produced. This path is continuous with constant length. The nth successive Fig. 101 curve of this procession has no unique slope at the set of points whose coordinates, measured from A, are of the form AB K 71i , K = 1,..., n.

(d) The "snowflake" (Von Koch curve) is the limit of the procession shown.*(Each side of the original

Cv") Cy-

Fig. 102

equilateral triangle is trisected, the middle segment discarded and an external equilateral triangle built there). The limiting curve has finite area, infinite lerzlh, and no derivative anywhere. The determination of length and area are good exercises in numerical series.

This procession is the one devised by Boltzmann to visualize certain theorems in the theory of gases. See Math. Annalen, 50(1898). FUNCTIONS WITH DISCONTINUOUS PROPERTIES 107 (e) The SterpInski urv( Is the limit !.he f, has 0 area, Inr.!nite

Fig. 10!.

n,L., derivative anywhere, and passes through every p,7:int w!thin the orivinal square.

(f) The WeLer;trass func tion y = bncos(annx),

where a is an cdd positive Inter,er, b a positive ::nstant less than unity, althcugh contInuous has no derivativp

at, >

BIBLIOGRAPHY

Edwards, J.: Calculus, Macmillan (1892) 225. Hardy, G. H.: Pure MathematicsMacmillan (1933) 162 ff. KasnPr and Newman: Mathematics and Imagination,Simon and Schuster (194o). Oswood, W. F.: Real Variables, Stechert(19)8) Chap. III. Plerpont, j.: Real Variables, Ginn and Co.(1912) Chap. XIV. SI-ph:haw:, H.: Maaematical 5napsh0t, Stechert(1938) 60. GLISSETTES

HISTORY: The idea of Glissettes in somewhat elementary form was known to the ancient Greeks. (For example, the Trammel of Archimedes, the Conchold of Nicomedes.) A systematic study, however, was not made until 1869 when Be.sant published a short tract on the matter.

1. DEFINITION: A Glissette is the locus of a point - or the envelope of a curve - carried by a curve which slides between given curves. An interesting and related Glissette is that generated by a curve always tangent at a fixed point of a given curv'. (See 6b and 6c below.)

2. SOME EXAMPLES: (a) The Glissette of the vertex P of a rigid angle whose sides slide upon two fixed points A and B is an arc of a circle. Furthermore, since Ptravels on a circle, any point Q of AP describes a Limacon. (See 4).

Fig. 104

(b) Trammel of Archimedes. A rod AS of fixed length slides withits ends upon two fixed perpendicularlines. GLISSETTES 109 1. The Glissette of any point P of the rod(or any point rigidlyattached) is an Ellipse. 2. The envelope Glissette ofthe rod itself is the Astroid. (See Envelopes,3a,) (c) If a point A of a rod, whichpasses always through a fixed point 0, movesalong a given curve r = f(U), the Olis- sette of a point P of the rod k units distant from A is the Conchoid r = f(0) k of the given curve. (See Moritz, R, E., U. of Wash. Pub. 1923, for pictures of many Fig. 105 varieties of this family, where the base curve is r =cos(29)].

3. THE POINT GLISSETTE OF A CURVESLIDING BETWEEN TWO LINES AT RIGHT ANGLES (THE x,yAXES):

If the curve be givenby p = f(T) referred to thecar- tied point P, then y = p = f(T) and x = + are parametric equations of the Glissette traced by P. Por example, the Astroid p = sin 2T, referred toits center, has theGlissette x = sin 2T, y = - sin 2, (a segment of x + y = 0) err the locusof its center as it slides between the x and y axes, 110 GLISSErrES

h. A TRTANAR TUCHINC; FIXEL ()flits:LES:

! v-1 a,-.!vpr: irl- aL,:ie ABC, w. \.:!1 :; :'i xod 10:3vi. th ,1t.or... X, Y. As this tv!ano,..le moves, lines XA' and YA' drawn parallel to the sides are lines fixed to the triangle. Let the circle described by A' meet the parallel to BC through A'it' D. Then angle PA'X = angle A'B'C angle AEC, all constant, and thus D is a fixed point of the circle. The perpendicu- Fig. 107 lar DP from D to BC is the altitude of the invariable triangle A'B'C' and thus BC touches the circle with that altitude as radius and center D.* The pc,Int Olissottos (For example, any point Por A'C' or thf m le are L:ma..ons. (See Troehoid:; 3th)

5. GENERAL THEOREM: Any mtion of a confkuration its _Aisne can be reprocented 11 the rolling of A certain doterminate curve on another' determinate curve. This reduces the problem of Glis- setten to that of Roulettes. A simple illustration is the trammel AB sliding upon two perpendicular lines. the instantaneous center of rotation of AB, lies always on the fixed circle with center 0 and radius AB. This point also lies on the circle having AB as diameter - a circle car-

Flm. ,;3 rted with AB. The action then

..,is' the sties or Iry pf,lywon envelope cirf:lea If two sides tWx.'n c'n!iPq or ;.sso througL two fixed points. Phin enters the iesign of a rotor, n convex c,irve which reenina tnngent to all 0H 6rors fired nlygon while the curve Is rotated.See ';ollbetw, X.: v950 W-ht.r. GLISSETTES is as if this smaller circle were rolling internally upon a fixed circle twIL'e as llrge. Hence, any point of AB describes an Eliipse awl the envelope or AB is the Astroid.

GENERAL ITEMS: (a) A Parabola slides on the x,y axes. The locus c: the vertex is: A x2y2(x Y2 3a2) = a6 ;

the focus is: 1 x2y2 = a2(x2 + y2).

(b) The path of the center of an Ellipse touching a straight line always at the same point is

x2y2 (a2 y2)(y2 b2).

(c) A Parabola slides on a straight line touching it at a fixed point of the line. The locus of the focus is an Hyperbola.

(d) The bar APB, with PA = a, PB = b, moves with its ends on a simple closed curve. The difference between the area of the curve and the area of the locus described by P is

nab.

Fig. 109 112 GLISSETTES (e) The vertex of a carpenter's square moves upon a circle wh!le one arm passes through a fixed point F. The envelope of the other arm is a conic with F as focus, (Hyperbola if F is outside the circle. Ellipse if inside, Parabola if the circle be replaced by a line.)

(See Conics 16.)

Yig. 110

BIBLIOGRAPHY

American Mathematical Monthly: v 52, 384; v 55, 393.402. Besant, W. H.: Roulettes and Glissettes, London (1870). Encyclopaedia Britannica, 14th Ed., "Curves, Special." Walker, G.: National Mathematics Magazine, 12,13(1937-8, 1938-0, HYPERBOLIC FUNCTIONS

HISTORY: or disputed origin; either by Mayeror by Riccati in the 18th century; elaboratedupon by Lambert (who provel the irrationality of n).Further investigated by Gudermann (1798-1851), a teacher orWeierstrass. He complied 7-place tables for logarithms of thehyperbolic functions in 1532.

1. DESCRIPTION: These functionsare defined as follows:

(ex ex) (ex + ex) 1

Binh x - , cosh x = ' . V(1 + sinh24, 2 2

sinh x 1 tanh x = coth x = cosh x tanh x

1 1 sech x = , each x = cosh x el x

y= cash r --- y sech x -

Fig. 111 114 HYPERBOLIC FUNCTIONS 2. INTERRELATIONS:

(a) [liv,,r.sk- Rlat!_2r1r2: arc sInh x = ln(x +vix2 + i), x2 < .;

arc cosh x = ln(x +Ix - I), x7 1;

2 arc tanh x =(1)' r(i+x) 2'11(1-x)" x < 1; /1% r) arc coth x =Tlni(x(x+101, x2 > I;

1n(- 1 2 arc sech x= + ).2-1) , 0< x < 1; X - X

1 1 2 arc csch x= 1n-1 + 1 x > 0. / i X X2 (

(b)Identities: cosh2r sinh2x = 1; sech2x = 1 - tanh2x;

csch2x = coth2x -1; sinh(x + y) = sinh x.cosh y 4 cosh vsinh y; cosh(x + y) = cosh x'cosh y + sinh x'sinh y; sinh 2x = 2sinh xcosh x; cosh 2x = cosh2x + sin2x;

tanh x f tanh y X f COS11. 1. Yl -1 + tanh x tanh y ,s'-- 2 = 2

y ip0Sh X + 1 cosh . 2 2

sinh x + sinh y = 2sinh cosh 1:1 2

cosh x + cosh y = 2cosh I±X cosh 2 2 ' sinh 3x = 4sinh3x + 3sinh x; cosh 3x = 4cosh3x 3cash x; %k (sinh x + cosh x) = sinh kx + cosh kx. HYPERBOLICFUNCTIONS 115 (c) Differentials and Integrals:

d( Binh x) = cosh xdx; ftanh x dx = In cosh x;

d( cosh x) = sinh k dx; coth x dx = In I einh x I :

d( tanh x) = sech2x dx; sech x dx = arc tan( sinh xl =

gd x ; x d( coth x) - -cach2x dx; cech x dx = lnitanh(-2 )1 3

d( sech x) = -sech xtanh x dx;

d(oeoh x) = -cech xcoth x dx;

dx d( arc e inh t ; d( arc cosh = + *1 417:37

dx d(arc tanh x) = = d( arc coth x),( in different inter- (1 x vale);

dx tdx d( arc each x) = + ; d( arc cech x) = x I x2 x

(called the "gudermannian") x = sec y dy = In' eeo y + tan y I . 0

3. ATTACHMENT TO THE RECTANGULAR HYPERBOLA: A comparison with the trigonometric (circular) functions is as follows.

Fig. 112 Irok.rarnrr,

116 HYPERBOLIC FUNCTIONS

For the shaded sectors ( A):

x = ucos t [x . :ocosh t

. y = asin t y :-, rosinh t 1 2 di\ (-2 )pdO,

Y u . arc tan = t, U =arc tun arc tan(tanh t),

dt dO = dt. dU - (coset + oinhat)

But

p2 . a2( cos2t + sin2t) = a2, p2= a2(cobh2t + sinh2t),

and thus t. t 2t 2 2dt a 1ir2 at A.(1)jfa . A = (-) adt = --- . 2 -2- ' 2 2 0 0 In Other case:

2A "L. a2

. acosEA x = a.cosh3-1. a2 a2

2A . 2A y = asin -7 y = wsnni 7 . a i'x

Thus the Hyperbolic functions are attached to the Rectangular Hyperola in the same manner that the trigonometrIl funolons are attached to the circle.

4. ANALTIICAL REW ::ONS WITH THE TRIGONOMETRIC FUNCTIONS:

The Euler + "sin x; e"x = cos( lx) + 181n( Ix);

(,"1"4 . -os x isln x;ex = cos( lx) ix); produce;

K; , os I; cosh x = cos( ix);

Binh ( i - t sin xi Binh x = -1.0In(IX);

from uhich relat Iona a abe derived. HYPERBOLIC FUNCTIONS 117 5. SERIES REPRESENTATIONS:

2k-1 sink x X".2 < cc; (2k-1): '

cc 21: 2 cosh x =2---- x < ( 21C ) 0

3 a 7 X 2x 17X 2 e + ; tanh x = X 3 15 + .4., X < 7

3 5 7 1 x 2x X 2 coth x=-+ 2( x 3 45 945 4725 4. A

1 2 5 4 61 .3 1385 8 2 n2 sech x ... 1 - x + x . grX + x ..., 8: , X ;

1 x 7x3 31x5 2 csch x = xx2 < n ; x 6 360 15.120+ 46.- .

1 x3 1.3 x5 1.3.5 x7 arc sinh x = x - + +.... x< 1 3 2.4 5 2.4.6 7

1 1 1.3 1 1.3.5 1 . in 2x + so., X > 1 ; 2 2x2 2.4 4x4 2.4.6 6x6

1 1 1.3 1 1.3.5 1 arc cosh x = In 2x - - +se., X >1 ; 2 2x2 2.4 4x4 2.4.6 6x6

Q 1 x2k arc tanh x = 2k-1 1

1 3 1 61 7 gdx= arc tan (sinh x) =x-gx + x - X + 24 540

6. APPLICATIONS: (a) y m a'cosh the Catenary, is the form of a flexible chain hanging from two supports. (b) These functions play a dominant role in electrical communication circuits. For example, the engineer prefers the convenient hyperbolic form over the exponential form of the solutions of certain types of problems in transmission. The voltage V (or current I) satisfies the differential equation

it? 118 HYPERBOLIC FUNCTIONS d2V --7 = zyV, dx where x is distance along the line, y the unit shunt admittance, and z the series impedance. The solutions

V = Vricosh x yz + x "Yr,

gives the voltage in terms of voltage and current at the receiving end. (c) Mapping: In the general problem of conformal world maps, hyperbolic functions enter significantly. For instance, in Mercator's (1512-1594) projection of the sphere onto its tangent cylinder with the N-S line as axis, x = 0, y = gd y, where (x,y) is the projection of the point on the sphere whose latitude and longitude are y and 0, respectively. Along a rhumb line, p =gd(Otan a + b), where m is the inclination of a straight course(line) on the map.

BIBLIOGRAPHY

Kennelly, A. E.: Applic. of 1122. Functions to Elec. gm. Problems, McGraw-Hill (1912). Merriman and Woodward: Higher Mathematics, John Wiley (1896) 107 ff. Slater, J. C.: Microwave Transmission, McGraw-Hill (1942) 8 ff. Ware and Reed: Communication Circuits, John Wiley(1942) 52 ff. INSTANTANEOUS CENTER OF ROTATION and THE CONSTRUCTION OF SOME TANGENTS

1. DEFINITION: A rigid body moving in any manner whatso- ever In a plane has an instantaneous center of rotation. This center may be located if the direction of motion or any two points A, B of the body are known. Let their respective velocities be VI and V2. Draw the perpendiculars to V1 and V2 at A and B. The center of rotation is their point of intersection H. For, no point of HA can move toward A or H (since the body is rigid) and thus all points must move parallel to V. Fig. 113 Similarly, all points of HB move parallel to V2. But the point H cannot move parallel to both VI and V2 and so must be at rest.

2. CENTRODE: If two points of a rigid body move on known curves, the instantaneous center of rotation of any point P of the body is H, the intersection of VH the normals to the two curves. 1-\ The locus of the point H is / called the Centrode. (Chasles)

Pig. 114 120 INSTANTANEOUS CENTER OF ROTATION 3. EXAMPLES: (a) The Ellipse is produced by the Trammel of Archi- medes. The extremities A, B of a rod move along two perpendicular lines. The path of any point P of the rod is an Ellipse.* AH and BH are normals to the directions of A and B and thus H is the center of rotation of any point of the rod. HP is normal to the path of P and its perpendicular PT is the tangent. (See Fig. 115 Trochoids, 3n.) (b) The Concholdt is the path of Pi and P2 where A, the midpoint of the constant distance P1P2, moves along the fixed line and PiP2 (extended) passes through the fixed point 0. The point of P1P2 passing through 0 has the direction of PiP2. Thus the perpendiculars OH and AH locate H the center of rotation. The perpendiculars to Fig. 116

* The path of P is an Ellipse if A and B move along any two inter. secting lines. (For a more general definition, see Oonohold, 10 DOTAIVENNECKNICENTEROFROTATION 121 Pill and PRH at PI and P2respectively, are tangents to the curve. (c) For the Limacon, Bmoves alung the circle while OBP rotates about 0. At any instant B moves normal to N,c) the radius BA while the N point on CP at 0 moves in

-",..:.-11;-..---;:\ the direction OP. The _It-.-:_--- -:--- center of rotation Is , - -;,(6.j , thus H (a point of the ,- circle) and the tangent / 1--' /, . \ 6 .<2- to the Limacon described .,,!.N .-- 'i. 1 by P is perpendicular to n 7"-- . ,,,-11; 'Ago PH. 7' /7 \ \ \ /7 / . a / \, /7 /

-... _....---

Vig. 117

(d) The Isoptic ofa curve 1s the locus of the intersection of two tangents whichmeet at si constant angle. Ii' these tangentsmeet the curve In A and B, ro-mals there to the givencurve meet in R. This Is the center of rotation ofany paint of the rigid body formed by the constantangle. Thus HP Is normal to the path or P. For example, (see Glissettes,4) the locus of the vertex of a triangle, two of whose sides touch fixed otr,clec, 18 a Limacon. Normals to these tangents pass through the centers of the circles and make acon- stant angle with each other. They meet at It, ,.. / the center 1 of rotation, and the locus of H is accordingly a circle / 4N through the centers of the two given circles. Pig. 118 122 , INSTANTANEOUS CENTER OF ROTATION (e) The point Glissette of a curve is the locus of P, a point rigidly attached to the curve, as that curve slides on given fixed curves. If the points of tangency are A and B, the normals to the fixed curves there meet in H, the center of rotation. Thus HP is normal to the path of / (P.

Fig. 119

(f) Trochoidal curves are generated by a point P rigidly attached to a curve that ,---- rolls upon a fixed curve. The point of tangency H is the center.

;

1 of rotation and HP is normal to (7 the path of P. This is particu- \\\ larly useful in the trochoids of H a circlet the Epi- andHypo- cycloids and the ordinary Cycloid.

Fig. 120

BIBLIOGRAPHY

Chasles, M.: Histoire de la04om6trie, Bruxelles (1881) 548. Keown and Faireus Mechanism,McGraw-Hill (1931) Chap. V. MiewenglOwski, B.: Cours de06omkrie Analytique, (Paris) (1894) 347 ff. Williamson, B.: Calculus, Longmans,Green (1895) 359. INTRINSIC EQUATIONS

INTRODUCTION. The choice of reference system for a particular curve may be dictated by its physical characteristics or by the particular type of information desired from Its properties. Thus, a system of rectangular coordinates will he selected for curves in which slope i3 of primary importance. Curves which exhibit a central property - physical or geometrical - with respect to a point will be expressed in a polar system with the central point as pole. This is well illustrated in situations involving action under a central force: the path of the earth about the sun for example. Again, if an outstanding feature is the distance from a fixed point upon the tangent to a curve - as in the general problem of Caustics - a system of pedal coordinates will be selected. The equations of curves in each of these systems, however, are for the most part "local" in character and are altered by certain transformations. Let a transformation (within a particular system or from system to system) be such that the measures of length and angle are preserved. Then area, arc length, curvature, number of singular points, etc., will be invariants. If a curve can be properly defined in terms of these invariants its equation would be intrinsic in character and would ex- press qualities of the curve which would not change from system to system. Two such characterizations are given here. One, relating arc length and tangential angle, was introduced by Whewell; the other, connecting arc length and curvature, by Cesiro. 124 iNTRINSIC EQUATIONS 1. THE WHEWELL EQUATION: The Whewell equation is that connecting arc length s and tangential angle y, where y is measured from the tangent to the curve at the initial point of the arc. It will be convenient here to take this tangent

o, /0 x as the x-axis or, in polar coordinates, the initial line. Examples

Fig. 121 follow.

(a) Consider the Catenary: y = a.cosh(i).

Here y' = sinh(i) = tan y; ds2 =(1 + sinh2(8))dx2.

z ix. ,x. Thus s = cosh( -)dx = a.sinhka ), and sx a.tan a Is fo (This relation is, of course, a direct consequence of the physical definition of the curve.)

(b) Consider the Cardioid: r = 2a(1 - cos e).

(1- cso 0) Here tan y tan(11) and thusy = sin0 2

However, y = y + 0, and thus y,m

The arc length: ds2 = 8a2(1 - cos e)de2

0 s -8a'cos(-2 ) -8a.cos(1) . 3 /NTFUNSIC EQUATIONS 125 The equation of an involute of a given curve is

obtained directly from the Whewell equation by inte- . gratIon. example, the circle: a = a.y

has for an involute: s_ 2 ' the constant of integration determined conveniently.

NOTE: The Inclination y depends of course upon the tangent to the curve at the selected point from which s is measured. If this point were selected where the tangent is perpendicular to the original chdtce, the Whewell equation would involve the co- function of y. Thus, for example, the Cardioid may be given by eitheo 'of the equations: s = k.cos(1) or s =k.cin(2).

2. THE CESARO EQUATION: The Cesttro equation relates arc length and radius of curvature. Such equations are definitive and fellow directly from the Whewell equations. For example, consider the general family of Cycloidal curves: s = asin bc.

Here ds R= = are. cos h c. dy R2 bs.sa a2bs, Accordingly, 4 126 INTRINSIC EQUATIONS 3. INTRINSIC EQUATIONS OF SOME CURVES:

Curve Whewell Equation Ceettro Equation r Actroid a = ElC08 2y 46 + R2 = 14a2

Cardioid 8 = coe(1) 62 + 9R2 = a2

2 2 Catenary a = atan y 6 + a. aR

Circle 6 = y R = a ------7 Clem) id a = a( sec y - 1) 729( 6 + a) =a21 9( 6 +a R2]3 2 ,2 2 Cycloid 6 = wen y 8 + n sia

81) Deltoid 2 = -- coo 3, 9a2 + R2 = 64b2 3

Epi- and * R2 0 . a2 a2b2 s = wain by Hypo-cyclolds

Equiangular a =aqe"- 1) m(6 + a) R Spiral

Involute of 2 6 =2:2! 2an = R Circle 2

es Nephroid a = 6bsin2 4R2 36b2 2 2 2 2 213/a Tractrix a a 01n 80C q, a+ R = a46 ----...

b < 1Epi.

b = 1Ordinary.

b > 1Hypo.

B/BLIOORAPHY

Boole, 0.: Differential Equations,London, 263. CambridgePhilosophical Transactions: VIII 689) IX 150. Edwards, J.: Calculus, Macmillan(1892). INVERSION

HISTORY: Geometrical inversion seems to be due to Steiner ("the greatest geometer since Apollonius") who indicated a knowledge of the subject in 1824. He was closely followed by Quetelet (1825) who gave some examples. Apparently independently discovered by Bellavitis in 1836, by Stubbs and Ingram in 1842-3, and by Lord Kelvin in 1845. The latter employed the idea with con- spicuous success in his electrical researches.

1. DEFINITION: Corsider the circle with center 0 and radius k. Two poilts A and A, collinear with 0, are mutually inverse with respect to this circle if :

(0A)(64= k2. In polar coordinates with 0 as pole, this relation is r.p =k2 in rectangular coordinates: // c)K

Fig. 122

k2x ley xi 777 / Y1 xm yg (If this product is negative, the pointsare negatively inverse and lie on opposite sides of 0.)

Two at:I'VE:3 are mutually inverse if every point of each has an inverse belonging to the other. 128 INVERSION 2. CONSTRUCTION OF INVERSE POINTS:

A 0

Fig. 123

For the point A inverse to Compass Construction: Draw A, draw the tangent AP, the circle through 0 with then from P the perpendicu- center at A, meeting the lar to OA. From similar circle of inversion in P, Q. right triangles Circles with centers P and Q through 0 meet in A. (For OA k or (0A)(0A) = k2. proof, consider the similar k=OA isosceles triangles OAP and PdA.)

3. PROPERTIES: (a) As A approaches 0 the distance OA increases in- definitely. (b) Points of the circle of inversion are invariant. (c) Circles orthogonal to the circle of inversion are invariant. (d) Angles between two curves are preserved in magni- tude but reversed in direction. (e) Circles:

r2 Ar.cose B.r.sine C = 0 = x2+ y24. Ax +By+C invert (by rp = 1) into the circles:

1 +lop.cose +B.() 'sine Cp2=C(x2+y2)+ Ax By + 1= 0 INVERSION 129 unless C = 0 (a circle throughthe origin) in which case the circle inverts into theLine: 1 + A.p.cos0 + B.p.sin0= 1 + Ax + By = 0. (f) Lines through the origin: Ax + By = 0 = A.cos0 + 'Paine are unaltered.

(g) Asymptotes ofa curve invert into tangents to the inverse curve at the origin.

4. SOME INVERSIONS: (k= 1) (a) With center ofinversion at Its vertex, a Parabola in- 1Y verts into the Cissoid of .--.

Dioeles. // i \ , \ x y2= hx*--* (x2Y' 7 = hx, ( l / a 2 hx \`. ,/ or \.. Y° (1- hx)

Fig. 124

(b) With center of inversion at a vertex, the Rectangular Hyperbola Inverts into the ordinary Strophoid.

x2 - y2 + tax= 046-0x2 - y2 + 2ax(x2 + y2) = 3,

or 2 2 1 2ax Y '1- 2ax

Fig, 125 130 INVERSION (c) With center of inversionat its center, the Rectangular Hyperoolainverts into a Lemniscate.

r2cos20 1. p2 m cos 20.

r /

1: x

Fig. 126

Conics (d) With center ofinversion at a focus, the invert into Limacons.

1 = a +b'cos Q. r (a, + bcos 0)

a Jr

Fig. 127 INVERSION 131 (e) With center of inversionat their center, confocal Central Conicsinvert into a family of ovalsand "figures eight."

x + 1 (a2+ X) (b +X) 2 X2 efe gY

(x2 y2)2.

Fig. 128

5. KECHANICALINVERSQRS:

Fig. 129

Parallel- The PeAuPellier Cell(1860,1 The Hart Crossed ogram carries fourcollinear the first mechanical 132 INVERSION inversor, is formed of two points 0, P, Q, R taken on rhombuses as shown. Its a line parallel to the appearance ended a long bases AD and BC.* Draw the search for a machine to circle thrutwh D, A, P, convert circular motion and Q meeting AB in P. By into linear motion, a the secant property of problem that was almost circles, unanimously agreed insol- able. For the inversive (BF)(BA)= (BP)(BD). property, draw the circle Here, the distances BA, BP, through P with center A. and BD are constant and Then, by the secant prop- thus BF is constant. Ac- erty of circles, cordingly, as the mechanism is ueformed, F is a fixed (OP)(0Q)= (0D)(0C) point of AB. Again, (a-b)(a+b) = a2- b2. (013)(0Q) = (OF)(0A)= con- Moreover, stant (P0)(PR) .-(0P)(00 =b2-a2 by virtue of the foregoing. if directions be assigned. Thus the Hart Cell of four bars is equivalent to the Peaucellier arrangement of eight bars.

For line motion, an extra bar is added to each mechanism to describe a circle through the fixed point (the center of inversion) as shown in Fig. 130.

Pig. 130

These remain collinear ae the linkage is deformed. INVERSION 133 In each mechanism, the line generatedis perpendicular to the line of fixeu points.

6. Since the Inverse k cf 7 lies 'in thepolar of I, the subject of inversion 's that of poles and polars, with respect to the given circle. The points 0, P, A, and A form an harmonic set - that is, A and X divide the distance OP in "extreme and mean ratio". A generalization of inversion leads to the theory of polars with respect to curves other than the circle, viz., conics. (See Conics, 6 ff.) Fig. 131

7. The process of inversionforms an expeditious method of solving a variety ofproblems. For example, the celebrated problem of Apollonius(see Circles) is to construct a circle tangent to three given circles. If the given circles do not intersect, each radius is inc..eased by a length a so that two are tangent. This point of tangency is taken as center of inversion so that the inverted configuration is composed of two parallel linesand a circle. Thecircle tangent to these three elements is easily obtained by straightedge ----- and compass. Theinverse (with respect to the same Fig. 132 circle of inversion) of this circle followed by analteration of 'ts radius by the. length a is therequired circle. 1,4 INVERSION 8. Inversion is a helpful means of generating theorems or geometrical properties which are otherwise not readily obtainable. For example, consider the elementary theorem: . "If two opposite angles of a quadrilateral OABC are supplemen- tary it is cyclic." Let this configuration be inverted with re- A :, spect to 0, sending A, B, C into X, S, t and their circumcircle 4 into the line A. Obviously, .5 lies on this line. If B be allowed to move upon the circle, t moves upon a line. Thus "The locus of the intersection Fig. 133 of circle.) on the fixed points 0,X and 0,C meeting at a constant angle (here n -0) is the line AC."

BIBLIOGRAPHY

Adler, A.: Geometrischen Konstruktionen, Leipsig (1906) 37 ff. Courant and Robbins: What is Mathematics? Oxford (1941) 158. Daus, P. H.: College Geometry, Prentice-Hall (1941) Chap. 3. Johnson, R. A.: Modern Geometry, Houghton-Mifflin (1929) 43 ff. Shively, L. S.: Modern Geometry, John Wiley (1939) 80. Yates, R.C.: Geometrical Tools, Educational Publishers, St. Louis (1949). INVOLUTES

HISTORY: The Involute of a Circle was discussed and utilized by Huygens in 1693 in connection with his study of clocks without pendulums for service on ships of the sea.

1. DESCRIPTION: An involute of a curve is the ruulette of a selected point on a line that rolls(as a tangent upon the curve. Or, it is the path of a point of a string tautly unwound from the curve. Two facts are evident at once: since the line is at any point normal to the involute, a].1 involutes of a given curve are parallel to each other, Fig. 134(a); further, the evclute of a curve is the envelope of its normals.

(a) Fig. 134 (b)

The details that follow pertain only to the Involute of a Circle, Fig. 134(b), a curve interesting for itsappli- cations. 136 INVOLUTES 2. EQUATIONS:

[x = a(cost + t.sint)

y = a(sint - t.cost) .

p2= r2 - as (withrespect to 0). zr2 TEs2=/10 + a.arc cos (11)

2a= ace = ate 2aa (= a2t21.

). METRICAL PROPERTIES:

A= (bounded by OA, OP, AP). 26a

4. GENERAL ITEMS:

(a) Its normal is tangent tothe circle. (b) It is the locus of the poleof an Equiangular spiral rolling on a circle concentricwith the base circle (Maxwell, 1849).

(d) It is the locus of theintersection of tangents drawn at the points where any ordinateto OA meets the circle and the corresponding cycloid having itsvertex at A.

(e) The limit of a succession ofinvolutes of any given curve is an Equiangular spiral. (See Spirals, Equiangular.)

(r) In 1891, the dome of the RoyalObservatory at Greenwich was constructed in the form of the surface of revolution generated by an arc ofan involute of a circle. (Mo. Notices Roy. Astr. Soc., v 51,p. 436.) (g) It is a specialcase of the Euler Spirals.

(h) The roulette of the center of theattached base circle, as the involute rolls on a line, is a parabola. INVOLUTES 137 (i) Its inverse with respect to the base circle is a spiral tractrl.x (a curve which in polar coordinates has constant tangent length). (j) It is used frequently in the desiol of cams. (k) Concerning its use in the construction of gear teeth, consider its generation by rolling a circle together with its plane along a line, Fig. 135. The path of a selected point P of the line on the moving plane is the involute of a circle. At any instant the center of rotation of P is the point C of the circle. Thus two circles with fixed centers could have their involutes tangent at P with this point of tangency always on the common internal tangent (the line Fig. 135 of action) ofhe two circles. Accordingly, a constant velocity ratio is transmitted and thefunda- mental law of gearing is satisfied. Advantages over the older form of cycloidal gear teethinclude: 1. velocity ratio unaffected by changing distance between centers, 2. corstant pressure or the axes, 3, siirle curvauure teeth(taus easier cut), 4, more uniform wear on the teeth.

BIBLIOGRAPHY

American Mathematical Monthly, v 28(1921) 328. Byerly, W. E.: Calculus, Ginn(1889) 133. Encyclopaedia Britannica, 14th Ed., under"Curves, Special". Huygens, C.: Works, laSoci4te Hollandaise des Sciences (1888) 514. Keown and Fairest Mechanism, McGraw-Hill(1931) 61, 125. ISOPTIC CURVES

HISTORY: The origin of the notion of isoptic curves is obscure. Among contributors to the subject will be found the names of Chasles on isoptics of Conics and Bpi- trochoids (1837) and la Hire on those of Cycloids (1704).

1. DESCRIPTION: The locus of the intersection or tangents to a curve (or curves) meeting at a constant angle % is the Isoptic of the given curve (or curves). If the constant angle be n/2, the isoptic is called the Orthoptic. Isoptic curves are in fact Oli'ssettes. A special case of Orthoptics is the Pedal of a curve with respect to a point. (A carpenter's square moves with one edge through the fixed point while the other edge forms a tangent to the curve).

2. ILLUSTRATION: It is well known that the Orthoptic of the Parabola is its directrix while those of the Central Conics are a pair of concentric Circles. These are immediate upon eliminating the parameter m between the equations in the sets of perpendicular tangents that follow: ISOPTIC CURVES 139

[y - mx ±Vg777= 0 my + x ±/a2t b2m2 = O. 2 y mx .1- pm =0 (The Orthoptic of the 2 Hyperbola is the circle in mx p = O. y through the foci of the corresponding Ellipse and vice versa.)

3. GENERAL IliMS: (a) The Orthoptic is the envelope of the circle on PQ as a diameter. (Fig. 1::,7) (b) The locus of the intersection of two perpendicular normals to a curve is the Orthoptic of its Evolute. (c) Tangent Construction: Fig. 137. Let the normals to the given cur.te at P and Q meet in H. This is thf instantaneous center of rotation of the rigid body formed by the constant angle at R. Thus HR is normal to the Isoptic generated by the point R.

4. EXAMPLES:

Given Curve Isoptic Curve

Cycloid Curtate or Prolate Cycloid Epicycloid Epitrochoid Sinusoidal Spiral Sinusoidal Spiral Two Circles Limacons (see Glissettes, 4) Parabola Hyperbola (same focus and directrix) 140 ISOPTIC CURVES

Given Curve Orthoptic Curve

Two Confecal Conics Concentric Circle

Hypocycloid -r 1( r = ( a21))421.111( - a.2b) 2 u) Deltoid Its Inscribed Circle

Cardioid A Circle and a Limacon

2 2 2 3 3 2 Astroid: x + y= a Quadrifollum: r2 = (--)008220

Sinusoidal Spiral: Sinusoidal Spiral: r - woosk (--0) where (n + 1) rn= an cooth k=

2 x3 y = x 72972= 180x - 16

3(x + y) =x3 8172(x2 + y2)-36(x2-2xy + 5y2) + 128 0 x2-2y 4a(x3 + y3) +

18a2ry-2ya4. 0 x + y + 2a = 0 Equiangular Spiral A Congruent Spiral*

NOTE: The m-Isoptic of the Parabola y2 = 4ax is the Hyperbola tan2m.(a + x12 y2- 4ax and those of the Ellipse and Hyperbola: (top and bottom signs resp.): y2 a2 b2)2 4(a2u.2 b2x2 a2b2). tan2 co(x2 - (these include the n - m Isoptics).

BIBLIOGRAPHY

Duporcq: L'Interm. d. Math. (1896) 291. Encyclopaedia Britannica: 14th Ed., "Curves, Special." Hilton, H.: Plane Algebraic Curves, Oxford (1932) 169.

*See An. Math. Monthly, v55, 317. KIEROID

HISTORY: This curve was devised by P. J.Kiernan In 1945 to establish a family relationship among theConchoid, the Cissoid, and the Strophoid.

1. DESCRIPTION: The center B or the circleof radius a moves along the line BA. 0 is afixed point, c units distant from AB. A secant is drawn through0 and D, the midpoint of the chord cut from the line DEwhich is parallel to AB and b units distant. The locusof PI and P2, points of intersection of OD and thecircle, is the Kieroid.

Fig. 138

The curve has a double point if c< a or a cusp if c = a. There are two asymptotes as shown. 142 KIEROID 2. SPECIAL CASES: Three special cases are of Importance:

If' U = 0, the If 1) . a, the If b a = -0 curve is a Con- curve ts a Cls- (points 0 and A chold of sold (plus an coincide), the Nicomedes. asymptote). curve is a Stroph- old (plus an asymptote).

Fig. 139

It is but an exercise to form the equations of these curves after suitable choice of reference axes. LEMNISCATE OF BERNOULLI

HISTORY: Discovered and discussed by JacquesBernoulli in 1694. Also studied by C. Maclaurin. JamesWatt (1784) of steam engine fame is responsible for thecrossed parallelor:Tam mechanism given at the end of this section. He used the device for approximateline motion - thereby reducing the height of his enginehouse by nine feet.

1. DESCRIPTION: The Lemniscate is a special It is the Cissoid of the Cassinian Curve, That is, circle of radius a/2 with it is the locus of a point respect to a point 0 dis- P the product of whose distant a 1/7272 units from tances from two fixed its center, points F1, F2 (the f 2a units apart is constant and equal to a2.

Fig. 140

(F1P)(FpP) = a2 r = OP = OB - OA = AB. A Point-wise Construction: Since sin r2 = sin () Let OX = a 47. Then) by the a v2/2 a secant property of the cir- 2 cle on FiF2 as diameter: r a.cos a =a Al- 2sin20), (XA)(X8) = a2. r2=a2.cos 20. Thus) take PIP at XB, F2P s XA, etc. 144 LEMNISCATE OF BERNOULLI 2. EQUATTaIS:

p`' = a'.Jr, 20, or = a2:::20,

(x2+ n?(x2- 12). (' = 2a2' :1, p3 = a26pe

3. METRICAL PROPERTIES: A = a". 1 16.; 161,65 L = 4a(1 ...) (elliptic). 25 26469 26466613 V (of r2= a2 co P 20 revolved about the polar axis) = 2ma2(2 - 72).

R =a2=r2' W = 2U )1, 2

4. GENERAL ITEMS: (a) It Ls the Pedal of a Rectwular Hyperbola with respect to its center. (b) It is the Inverse of a Rectanp:ular Hyperbola with respect to Its center. (The asymptotes of the Hyper- bola invert into ta:1ents to the Lemniscate at the

(c) It Is the Sinusoidal Spiral: rn = ancos nO for n = 2. (d) It is the locus of flex points of a family of confocal Cassinian Curves. (e) It is he enve]ope of circles with centers on a Rectangular Hyperbola which pass throull its center. LEMNISCATE OF BERNOULLI 145 (f) Tanient Contruct:on:

+ , the

normal ma'.:es 2t, with the L.adius vector and 5U with the pJlar a;;Is. The tan[.ent Is thus easily construe d.

(r) Radius of Curvature Fig. 141

a2 (Fir. 141) R = 7 . The 5r projection or R on the radius vector is 2 a R-oos 20 = H)-cos 2u = .50

Thus the Ur pendicular to the radius vector at Its trisection 2...).1nt farthest from 0 meets the normal in C, the center or curvature. (h) It is the path of a body acted upon bya central force varyin inversely as thee seventh power of the distance. (See Spirals 2r and 3rj---- 146 LEMNISCATE OF BERNOULLI (j) Generation Jay Linkages:

Fig. 142

OA =AB =la; BC CP set OC =a AS CD gm a ,C2-. AD = BC = a. Since angle BOP = always, P and 0 are midpoints of DC and AB, rasp. (N)2 (011)2 = r2 2 2 r m aCOS 20, 2a2 - 4a2sin20, (See Tools.) or r2 = 2e-cos 20. LEMNISCATE OF BERNOULLI b. 1117

BIBLIOGRAPHY

Encyclopaedia Britannica: 14th Ed., "Curves, Special." Hilton, H.: Plane Algebraic Curves, Oxford (1932). Phillips, A. W.: Linkwork for the Lemniscate, Am. J. Math. I (1878) 386. Wieleitner, H.: Spezielle ebcne Kurven, Leipsig (1908). Williamson, B.: Differential Calculus, Longmans, Green (1895). Yates, R.C.: Geometrical Tools, Educational Publishers, St. Louis (1949) 172. LIMACON OF PASCAL

HISTORY: D!scuvered ry Etienn,7- (rather ofBlalsc) Pascal and d:scusred 1.):; R:nerval in 1650.

1, DESCRIPTION: It is the. EpltrJch::d Tt Is the Conchold a c6.nerated a cir.:!le where the fixed attac:.ed a point is on the circle, circle roll ino. up ,)n an equal fixed circle,

Pig. 143

Cusp it 2a = k; Double point: 2a < k; Indentation: > k.

2. EQUATIONS: x = 4acost - lcos2t r = 2acose + k. y = liasint - ksin2t

(x2 y2 - 2ax)2 = k2(x2 + y2), (origin at singular point). LIMACON OF PASCAL 149 3. GENERAL ITEMS: (a) It is the Pedal of a circle with respect to any point. (If the point is on the circle, the pedal is the Cardioid.) (For a mechanical description, see Tools, p. 188.) (b) Its Evolute is the Catacaustic of a circle for any point source of light. (c) It is the Glissette of a selected point of an invariable triangle which slides between two fixed points. (d) The locus of any point rigidly attached to a constant angle whose sides touch two fixed circles is a pair of Limscons (see Glissettes 2a and 4). (e) It is the Inverse of a conic with respect to a focus. (The inverse of r = 2acose k is r(2acos0 k) = 0, an Ellipse, Parabola, or Hyper- bola according as 2a < k, 2a = k, 2a > k). (See Inversion 4d.) (f) It is a special Cartesian Oval. (g) It is part of the Orthoptic of a Cardioid. (h) It is the Trisectrix if k = a. The angle formed by the axis and the line Joining (a,o) with any point (r,$) of the curve is 3U. (Not to be confused with the Trisectrix of Maclaurin which resembles the Folium of Descartes.) 150 LIMACON OP PASCAL (i) Tangent Construction: The point A of the bar has Since T is the center of direction perpendicular to rotation of any point OA while the point of the rigidly attached to the bar at B has the direction rolling circle, TP is of the bar itself. The nor- normal to the path of P mals to these direct!.ons and its perpendicular at meet in H, A point, of the P is a tangent. circle. Accordingly, HP is normal to the path of P and its perpendicular there is a tangent to the curve.

(a) Fig. 144 (b)

(2a * 12 (j) Radius of Curvatures R = Oa k The center of curvature is at C, Pig.144(a). Draw HQ perpendicular to HP until it meetsAB in Q. C is the intersection of Q0 and HP. (k) Doutle veneration: (See Epicycloids.) It may also be generaLed by a point attached to acircle rolling internally (centers on the same side of the common tangent) to a fixed circle half the size of theroll.. ing circle. LIMACON OF PASCAL 151 (1) The Limacon may be generated by the followijle/ linkage: CDKF and COED are two similar (proportional) crossed parallelograms with points C and F fixed to the plane. CHM is a parallelogram and P is a point on the extension of JD. The action here is that produced by a circle with center D rolling upon an equal fixed circle whose center is C. The locus of P (or any point rigidly attached to JD) is a Limacon. (See an equivalent mechanism Fig, 145 under Cardiold.)

BIBLIOGRAPHY

Edwards, J.: Calculus, Macmillan (1892) 349. Salmon, G.s Higher Plane Curtest Dublin (1879). Wieleitner, H.: bezielle ebene Kurven, Leipsig (1908) 88. Yates) B.C.: Geometrical Tools, Educational Publishers St. Louis (1949) 182. N PHROID

HISTORY: Studied by Huens sndTschirnhausenabout 1679 in connection with the theory of caustics.Jacques Bernoulli in 1692 showed that the Nephroid isthe catacaustic of a cardioid for a luminous cusp.Double generation was first discovered by Daniel Bernoulliin 1725.

1. DESCRIPTION: The Nephroid is a 2-cuspedEpicycloid. The rolling circle may be one-half(a = 2b) or three- halves Oa 2b) the radius of the fixed circle.

Fig. 146

For this double generation, let thefixed circle have center 0 and radius OT = OE = a, andthe rolling circle center A' and radius A'T' = A'F' =a/2, the latter carrying the tracing point P. Draw ET',OT,F,and PT' to T Let D be the intersection of TO and FPand draw the circle on T, P, and D. This circleis tangent to the fixed circle since angle DPT =m/2. Now since PD is parallel to TIE, triangles OET' andOPT are isosceles and thus TD = 3a. NE PHROID 153 Furthermore, arc TT' = 2au and arc TIP= aft = arc T' X.

Thue ar:. TX !au arc 'PP. Accordingly, if P wore at-Ached to either rollim circle the one of radius a/2 or the one or radius 3a/2- the same Nephrold would be p:enerated.

2. EQUATIONS: (a = 2b) .

X . b(3cost -cos-..5t)

y . b(3sInt - stn3t) (x2 + y2- 4)3E12 = 108a4y2 .

s =6b'sin(2), 4R2 + s2= 36b2. 2

. 2 p 413'sit:(1). VP-4b2 +

(r/2)4=a4 . [sin4(2) + coA-91]. 2

x.cosy + y.sinp = 4b.sin(2).

3. METRICAL PROPERTIES: (a = 2b).

L = 24b. A = 12nb2. R -2 '

4. GENERAL ITEMS:

(a) It is the catacaustic or a Cardloid fora luminous cusp, (b) It Is the catacaustic of a Circle for a sot of parallel rays. (c) Its evolute is another Nephroid.

(d) It is the evolute of a Cayley Sextic (acurve parallel to the Nephroid). (e) It is the envelope of a diameter of the circle that generates a Cardioid. (f) Tangent Construction: Since T' (or T) is the instantaneous center of rotation of P, the normal is T'P and the tangent therefore PF (or Pa). (Fig. 151.) 154 NEPHROID

BIBLIOGRAPHY

Edwards, J.: Calculus, Macmillan(1892) 343 ff. Proctor, R. A.: A Treatise on the Cycloid(1878). Wieleitner, H.: qpezielle ebene Kurven, Leipsig(1908) 139 ff. PARALLEL CURVES

HISTORY: Leibnitt: was the first to consider Parallel Curves in l69-4, prompted no doubt by the Involutes of Huygens (1673).

1. DEFINITION: Let P be a variable point on a given curve. The locus of Q and Q', + k units distant from P measured along the normal, is a curve parallel to the given curve. There are two I; branches. For some values of k, a Parallel curve may not be unlike the given curve in appearance, but for other values of k it may be tutally dis- similar. Notice the paths of a pair Fig, 147 of wheels with the axle perpendicular to their planes.

2. GENERAL ITEMS: (a) Since Parallel Curves have common normals, they have a common Evolute. (b) The tangent to the given curve at P is parallel to the tangent at Q. A Parallel Curve then is the envelope of lines: ax + by + c kys7717, distant + k units from the tangent: ax + by + c 0 0 to the given curie. (c) A Parallel Curve is the envelope of circles of radius k whose centers lie on the given curve. This affords a rather effective means of sketching various parallel curves. 156 PARALLEL CURVES (d) All Involutes of a given

(fury° are parallel to earh

other (Fig. 1.48).

1 I

i I t e 1\Y

r /

Fig. 1.4.8

(e) The'differenee in lengths of two branches of a Parallel Curve is link.

3. SOME EXAMPLES: Illustrations selected from familiar curves follow. (a) Curves parallel to the Parabola are of the 6th degree; those parallel to the Central Conics are of the 8th degree. (See Salmon'sConics).

(b) The Astroid x = ahas parallel curves:. x2y2 2 a2) 4k21C- 127aXY9/0 X2+y2) - 18s2k 8k3 j2 = 0. _A 158 PARALLEL CURVES 4,A LINKAGE FOR CURVES PARALLEL TO THE ELLIPSE:

--- L P 0 ../1//4E

: 0

Fig. 150

A straight line mechanism is built from two proportional crossed parallelograms 001EDO and 00'FAO. The rhombus on OA and OH is completed to B. Since 001 (here the plane on which the motion takes place) always bisects angle AOH, the point B travels along the line 00'. (See Tools, p. 96.) Any point P then describes an El- lipse with semi-axes equal in length to OA + AP and PB. Since A moves on a circle with center 0, and B moves along the line 00', the instantaneous center of rotation of P is the intersection C of OA produced and the perpendicular to 00' at B. This point C'then lies on a circle with center 0 and radius twice OA. The "kite" CAPG is completed with AP 1. PG and CA sm CO. Two additional crossed parallelograms APMIA and PMARP are attached in order to have PM bisect angle APO and to insure that PM be always directed toward C. Thus PM is normal to the path of P and any point such as Q describes a curve parallel to the Ellipse. PARALLEL CURVES 159

BIBLIOGRAPHY

Dienger: Arch. der Math. IX (1847). Lqria, G.: Spezielle Algebraische and Transzendente ebene Kurven, Leipsig (1902). Salmon, G.: Conic Sections, Longmans, Green (1879) 337; Par. 372, Ex. 2. Wieleitner, H.: Spezielle ebene Kurven, Leipsig (1908). Yates, R. C.: American Mathametical Monthly (1938) 607. PEDAL CURVES

HISTORY: The idea of positive and negativepedal curves occurred first to Colin Maclaurin in1718; the name 'Pedal' is due to Terquem. The theory ofCaustic Curves includes Pedals in an important role: theorthotomic is an enlargement of the pedalof the reflecting curve with respect to the point source of light(Quetelet, 1822). (See Caustics.) The notion me I be enlarged uponto include loci formed by dropping perpendiculars upona line making a constant angle with the tangent -viz., pedals formed upon the normals to a curve.

1. DESCRIPTION: The locus C1, Fig.151(a), of the foot of the perpendicular from a fixedpoint P (the Pedal Point) upon the tangent to a given curveC is the First Positive Pedal of C with respectto the fixed point. The given curve C is theFirst Negative Pedal of Cl.

(b) (a) Fig. 151

It is shown elsewhere(see Pedal Equations, 5) that the angleop between the tangentto a given curve and the radius vector r from thepedal point, Fig. 151(b), equals the correspondingangle for the Pedal Curve. Thus the tangent to the Pedalis also tangent to the circle on r as a diameter.Accordingly, the envelope of theee circles is the first positive22141. PEDAL CURVES 161 Conversely, the first negative Pedal is then the envelope of the line through a variable point of the curve perpendicular to the radius vector from the Pedal point.

2. RECTANGULAR EQUATIONS: If the given curve be f(x,y) = 0, the equation of the Pedal with respectto the origin is the result of eliminating m between the line: y = mx +k and its perpendicular from the origin; my + x= 0, where k is determined so that the line is tangent to thecurve. For example; The Pedal or the Parabola y2 = 2x with respect to its vertex (0,0) is 1 [y= mx + s 2m 2 2x or y a Cissoid. my + x = 0 2X 1 9

3. POLAR EQUATIONS; If (r0,00) are the coordinates of the foot of the perpendicular from the pole: dO tan = r(-c-17,), ro = .sin and *+(0- 00)=

v2 1 dr2 Thus =1+ . ro r de Among theserelat'ons,r,8 and y Fig, 152 may be eliminated to give the polar equation of the pedal curve with respect to the origin. For example, consider the Sinusoidal Spirals ,r1% r1 = ancos nu.* Differentiating: nk7R) = -ntan nO

= n4cot y; thus p =-+ ne. 2

1 1 Rectifiable when- is an integer. 162 PEDAL CURVES

But e eo + - eo -ne and thus 0 = 2 (n +l

Now ro = rsin rcos ne a acoe nOoos ne, (n+i)/n or ro oos ne aacos(144)/4[1*0

Thus, dropping subscripts, the first pedal with respect to the pole is:

r a Icos ni0 where ni a another. Sinusoidal Spiral. The iteration is clear.The kth positive pedal is thus

nk rnk acos nk0 wherenk

Many of the results given in thetable that follows can be read directly from this last equation.(See also Spirals 3, Pedal Equations6.) 4.PEDAL EQUATIONS OP PEDALS: Let the given curve be r f(p) and let pi denote the perpendicular from the origin upon the tangent to the pedal. Then (See Pedal Equations):

p2 rp. f(p)p.. f\ 7 Thus, replacing p and pi by their \ respective analogs r and p, the pedal equation of the pedal is:

f(rT.EI] Pig. 45 I r2 Thus conAidgr the circlerill = ap. Here f(i., .,Virand f(r) ms/(ark). Hence, the pedal eruation of its Pedal with respect to a point on the circle is

ra 4/7---ar)13 or Ira ape

a Cardioid.(See Pedal Equations, 6.) Equations of successive pedals areformed in similar fashion. PEDAL CURVES 163 5.SOME CURVES AND THEIR PEDALS:

Given Curve Pedal Point Pint Positive Pedal

Circle Any Point Limacon

Circle Point on Circle Cardioid

Parabola Vertex Cieeoid Tangent at Inoue Parabola Vertex See

Auxiliary Conics) Focus Central Conic Circle 16.

Central Conic Center r2 = A + Bcos20

Rectangular Hyperbola Center Lemniscate

Equiangular Spiral Pole Equiangular Spiral t, Cayley'sSextic Cardioid (pea . r3) Pole (Cusp) ( r4=ap3) 5 3 Lemniecate (pas . r3) Pole r isap

Cataoaustio of a Parabola for rays perpendicular to its Pole Parabola axle 30 rcos(-.) a a 3

Sinusoidal Spiral Pole Sinusoidal Spiral (rte ap) 2r . t asin20 (Quadri. Astroidtx4+yl.. e Center folium)

Parabola Foot of DireotrixRight Strophoid

Arb. Point of Strophoid Parabola Dirsotrix

Reflection of Trisectrix of Focus in Direc. Parabola Maclaurin trix

Cieeoid Ordinary Focus Cardioid

121. and HypocycloidaCenter Roeee 164 PEDAL CURVES (Table Continued)

Given Curve Pedal Point First Positive Pedal

Deltoid * Cusp Simple Folium

Deltoid Vertex Double Folium

Deltoid Center Trifolium

Involute of a Circle Center of Circle Archimedian Spiral

3 3 + y= a Origin 1 3 (x2+ y2)4=dece +y)

n+m r = xmyn = awn Origin am+n.(m+n)1114.n.co8mosinn0 mmnn n n X 7 )n/( n..I.) n/(n.q.) ()+ () = 1 +( by) r2 a b Origin y2)n/(n..1) (Lame Curve)

(which for n = 2 is anEllipse; for n =1/2 a Parabola).

Its pedal with -eepect to (b,0) has the equation:

[(x - b)2 + y2jfy2 + x(x-b))= 4a(x - b)1,

where x2 + y2 = gat is the circumcircle of the Deltoid.

6. MISCELLANEOUS ITEMS:

(a) The 4th negative pedal of theCardioiu with respect to its cusp is a Parabola.

(b) The 4th posit've pedal ofr cos(.-)0` )% = a with 9 respect to the pole is a Rectangular Hyperbola. 3 (c) RI(2r2- pR) = r where R, R' are radii of curvature of a curve and its Pedal at correspondingpoints. PEDAL CURVES 165

BIBLIOGRAPHY

Edwards, J.: Calculus, Macmillan (1892) 163 ff. Encyclopaedia Britannica: 14th Ed., under "Curves, Special." Hilton, H.: Plane his. Curves, Oxford (1932) 166 ff. Salmon, G.: Higher Plane Curves, Dublin (1879) 99 ff. Wieleitner; H.: Spezielle ebene Kurven, Leipsig (1908) 101 etc. Williamson, B.: Calculus, Longmans, Green (1895) 224 ff. PEDAL EQUATIONS

1. DEFINITION: Certain curves have simple equations when expressed in terms of a radius vector r from a selected fixed point and the perpendicular distance.2 upon the variable tangent to the curve. Such relations are called Pedal Equations.

2. PROM RECTANGULAR TO PEDAL EQUATION: If the given curve be in rectangular coordinates, the pedal equation may be established among the equations of the curve, its tangent, and the perpendicular from the selected point. That is, with

f(xo,Yo) = 0,

(fy)o(Y- Yo)+(fx)0(x- xo)= 0,

2 + Yfs P (: '2,r2=X024101 { ((f(x ,2)04. ( ; 2)] where the pedal point is taken as the Pig. 154 origin.

S. FROM POLAR TO PEDAL EQUATION:

Among the relations: r = f(0), p = raisin 111, tan yr= 7, where the selected point is the origin of coordinates, 0 and41 may be eliminated to produce the pedal equation. (For example, see 6.) PEDAL EQUATIONS 167 4. CURVATURE IN PEDAL COORDINATES: The expressionfor radius of curvature is strikingly simple:

:;,..01VE

:.,._:::46. ...:HVE

+30

Fig. 15,

dr2 r2d02 dO Since ds2 and tany = = r(71),

dO/ds = p /r2 t=Hit)=pilre( )/r and thus

Now p = r.sin y and dp'n (sin Odr + r(cos Ody,

or LIS a\rdrl+tr.:43\ ds ri'ds' ds"

Thus 1.12 /210.2) ds u' rdr' r . da dy d ©_ (.110.2 Accordingly,'K = or ds ds ds 'r"dr

r(SIS) dp

5. PEDAL EQUATIONS OF PEDALCURVES: Let the pedal equation of a given curve be r =f(p). If pi be the perpendicular upon the tangent to thefirst positive pedal of the given curve, then, since p makes anangle of

4 with the axis of coordinates, 2

tan 0 a /4) (see Fig. 155). 168 PEDAL EQUATIONS Now tan y(13) = resin 41.()(2)

and thus tan y = sin y. (t) = tan %P..

Accordingly, y = y and p2 = rpi. In this last relation, p and 131 play the samo roles as do r and p respectively for the given curve. Thus the pedal equation of the first positive pedal of r = f(p) is

1r2= pf(r) (. Equations of successive Pedal curves ere obtained in the same fashion.

6. EXAMPLES: The Sinusoidal Spirals are E= ansinnel. Here, 7 = tan ne = tan W.

Thus y = nO, a relation giving the construction of tangents to various curves of the family. ra+1 p = r.sin y = r.sin nO = a n /14.1 or a = r , the pedal equation of the given curve Special members of this family are included in the following table:

2 Pedal an r n rn= ansin ne Curve Equation (n+1)rn-q(r1+1)p= r2sin20 + a2= 0Reet.11y-perbola rp = a2 -r3/a2 -1 r.sin9+ a= 0 Line p= a 0, -1/2 2a r-1-cos 0 Parabola p2 = ar 2 ,r7 /E7 a, , 2, +1/2r=(-2 )(1-cos U) Cardioid pea = r3 khrel, 3

2 a +1 f0 wain 0 Circle pa. r

2 as +2 r2'= a2uin 20 Lomniooate a = '" 314 tSeo also Spirals, 3 and Pedal Curves, 3.) PEDAL EQUATIONS 169 Other curves and corresponding pedal equations are given:

PEDAL CURVE PEDAL EQUATION POINT

all( r2 ..p2 )2 2( r2+4a2 Parabola (12 = 4a) Vertex Itt ) (p2+4a2)

b22 2a Ellipse Focus -- =-- - 1 p r 2 ab ,2 Ellipse Center --r -r2+ =a2+ 0

132

b2 2a Hyperbola Focus -17017 +1

22 ab 2 2 = -a +b2 Hyperbola Center 2 - r P --, 2 2 * * Epi- and Hypocycloids Center p = Ar+ B 2 2 2 Astroid Center r + 21) = a

Equiangular (a) Spiral Pole p = risin a

8p2 9r2 (12 Deltoid Center ____,... A Cotes' Spirals Pole = + B p22 r2

zn = am()*(Sacchl pe(me.rem aem) m2.r2M+2 Pole 1854)

*m = 1:Archisedean Spiral; m = 2:Fermat's Spiral; m = -1:Hyperbolic Spiral; m = 2:Lituus.

** (a + 2b)2, A e B r -a A. 4b(a + b)

BIBLIOGRAPHY

Edwards, J.: Calculus, Macmillan (1892) 161. glicurc120111a Britannica, 14th Ed., under "Curves, Special." WieleitnPr, H.: Spezielle ebene Kurven(1908) under "Fuenpunkts kurven." Williamson, B.: Calculus, Longmans, Green(1895) 227 ff. PURSUIT CURVE

HISTORY: Credited by some to Leonardo da Vinci, it was probably flyst conceived and solved by Bouguer in 1732.

1. DESCRIPTION: One particle travels along a specified curve while another pursues it, its motion being always directed toward the first particle with related velocities. If the pursuing particle is assigfied coordinates (x,y) and there is a function g relating the two velocitiesds da. dt dt then the thr,e conditions

Fig. 156 f(4, OJ x) Y

If) among which 4, n(coordinates of the pursued pArUse) may be eliminated, are sufficient to prodUoe the dif. feretAlal equation of the curve or pursuit.

Re PECIAL CA: at the partiole pursued travel rroM rest at the x-iwis along the llne x a, Pig. Int Oho pursuer starts a, the sem* Oar from the origin with Velocity k times the forlt.P. Son

YI 09 ri y 40 (000xY4

ds k,00 or d-X2 + dy2 ka.de There follow dx2 + dy2 = k fdy - 50,0 + (a - x)W12 kg(a x)2(dyl)k

OP 1-1 yie e(,,,Ltd P (a differentiel equation solviAle by first setting y' = p), Its solutions are PURSUIT CURVE . 171 -i/k(k+i)/k kaIlk(a -x)(k-i)/k ka (a x) 2ka = + if k i 1; 1 k 1 + k 1 - k21 2, t 4ay (a -102 2a2 la a x a if k = 1. a The special case when k = 2 is the cubic with a loop: a(3y - 2a)2 = (a - x)(x + 2a)2.

3. GENERAL ITEMS: (a) A much more difficult problem than the special case given above is that where the pursuedparticle travels on a circle. It seems not to have been solved until 1921 (F. V. Morley and A. S. Hathaway). (b) There is an interesting case in which three dogs at the vertices of a triangle begin simultaneously to chase one another with equal velocities. The path of each dog is an Equiangular Spiral. (E. Lucas and H. Brocard, 1877). (c) Since the velocities of the two particles are given, the curves defined by the differential equation in (2) are all rectifiable. It is an interesting exercise to establish this from the differential equation.

BIBLIOGRAPHY

American Mathematical Monthly, v 28,(1921) 54, 91, 278. Cohen, A.: Differential Equations, D. C. Heath(1933) 173. Encyclopaedia Britannica, 14th Ed., under"Curves, Special." Johns Hopkins Univ. Circ., (1908) 135. Luterbacher, J.: Dissertation, Bern(1900). Mathematical Gazette (1930-1) 436. Nouv. Corresp. Math. v 3 (1877) 175, 280. RADIAL CURVES

HISTORY: The idea of Radial Curves apparently occurred flirt wTucker in 1864.

1. DEFINITION: Lines are drawn from a selected point 0 equal and parallel to the radii of curvature of a given curve. The locus of the end points of these lines is the Radial of the given curve.

2. ILLUSTRATIONS: (a) The radius of curvature of the Cycloid (Fig.

157(a) (see Cycloid) is (R has inclinationn -t= 0):

R =2(PH)

Thus, if the fixed point be taken at a cusp, the radial curve in polar coordinates Is;

1 r = 4asin(i)= 4asin 6 a circle or radius 2a.

(a) Fig. 15'1' (b) RADIAL CURVES 173 (b) The Equiangular Spiral s = a(en* - 1) Fig. 157(b) has R = ma.03(f. Thus, if 0 be the inclination of the radius of curvature, U = 2 + y, and

m.a.em(U 4/2) 1 is the polar equation of the Radial: another Equi- angular Spiral.

3. RADIAL CURVES OF THE CONICS:

Fie. 158

x3 = f k'(x2 + y2) (a2x2 +b2y2)3=a4b4(x2+ y2)2

(Ellipse : b2 > 0;

Hyperbo:ta: b2 < 0) .

4. GENERAL ITEMS: (a) The degree of the Radial of an algebraic curve is the same as that of the curve's Evolute. 174 RADIAL CURVES 5. EXAMPLES:

Curve Radial

Ordinary Catenary Kampyle of Eudoxus Catenary of Un.Str. Straight Line Tractrix Kappa Curve Cycloid Circle Epicycloid Roses Deltoid Trifolium Astroid Quadrifolium

BIBLIOGRAPHY

EncyclAPEwlia BrItaquiaa 14th Ed., "Curves, Special." Tucker: Proc. Lon. jowl. sop., 14 (1465). Wieleitner, H.: Saeliel-1e Otno Amtlyen, Lei.psig (1908) 362. ROULETTES

HISTORY: Besant in 1869 seems to have been the first to give any sort of systematic discussion of Roulettes although prviously, Myer (1525), D. Bernoulli, la Hire, Desargues, Leibnitz, Newton, Maxwell and others had made contributions in one form or another, particularly on the Cycloidal Curves.

1. GENERAL DISCUSSION: A Roulette is the path of a point - or the envelope of a line - attached to the plane of a curve which rolls upon a fixed curve (with obvious continuity conditions).

Fig. 159

Consider the Roulette of the point 0 attached to a curve which rolls upon a fixed curve referred to its tangent and normal at 01 as axes. Let 0 be originally at 01 and let T:(xl,y1) be the point of contact. Also let (u,v) be coordinates of T referred to the tangent and normal at 0; y and T1 be the angles of the normals as indicated. Then x = v'sin(Y + 411)- u.cos(cf +yi) - xi

y = -1vcos(T + yl) - U.sin(y + y1) + yi , [ 176 ROULETTES where all the quantities appear In the pie.ht member may be exp-.:,::!ed In terms o!' t!:0 arc! s. These then ar, lx.us 0. Tt Is not difficult to eneral:e for any cprrItA point. Familiar examples of Roulet;es or a point 'we the Cy,!lo:ds, the Trcchoids, and Involutes.

2. ROULETTES UPON A LINE: (a) Polar Equatftn: 1:ersider the Roulet'e renevated by the point Q attPched to the curve - 1'()), referred to Q as pole (with Q0I as initial line),, as it rolls upon the x-axis. Let P be the point of tan- ,.ncy and the point O o: the curve be oricinally at O. The insta!taus centr of rotation of Q is P and thus for the locus JV 0:

= cot y dx But tan T.r(ar)and ,dx, y = r.sin = rk--). ds

Thu , among the relations:

dx ,dx% r I( ) = rk--) y =rk) dy dr ' ds the quantities r,U may be el* Alated Lc, (q0.111 the rectangular equation of the path of Q. Fl. 160 For example, considpr, Fig. 161, the locus of the focus cam;' the Parabola rnlling upon a line: originally the tangent at Its vertox:

2a dx 1 -sin 0 dx = r =1 - Llitt 0 dy cos U Y r.ds ROULETTES

Fi1;. 161

From thev.e, r and U are eliminated to :,ive

x ads = dx or a s= dx = A a definitive property of the Catenary (See Catenary,

(b) P-,.dal Equation: If the rollirn! .curve is in the form p f(r) (with respect to Q), then p = QN y ,dx, = rk v and the vectarwular equation of theroulette lo,.aver. by

.S-111) dx For example, cow:Ider the li:mlette or the pedal point (here the center of the fixed circle) of the Cycloidal

2 = A2(rP a2)I where A . a 2b, and

B 4b(a + b), as curve rolls upon the x-axis (ori?,inallo, a cusp tangent) . 178 ROULETTES The Roulette is given by

Bye ds 2 = A2(y-D /'dx---) - a2] = A2y2(1 + y'2) a2A2.

From this 2adx 2ydy A TAT_ y2

and

= -,/ A2 -y Ax the constant of integration being discarded by choos- ing the fixed tangent. Thus the Roulette is e y2 ax2 A4I

an Ellipse. As a particular case, Fig. 1b2, the Cardiold has a = b, and the Roulette of its pedal point Is 2 X + 9y2 = 81a2.

Fig. 162 ROULETTES 179 The Card old rolls on "top" of the line until the cusp tolJhes, then upon the "bottom" in the reverse direction. (c) Elegant theorems due to Steiner connect the areas and lengths of Roulettes and Pedal Curves: I. Lett point rigidly attached to a closed curve rolling upon a line generate a Roulette through one revolution of the curve. The area between Roulette and line Is double the area of the Pedal of the rolling curve with respect co the generating point, For example The area under one arch of the Ordinary Cycloid generated by a circle of radius a is 3na2j the area 'or the Cardioid formed as the Pedal of this circle with respect to a point on the circle is 2 22A_ 2 The Pedal of an Ellipse with respect to a focus is the ci:cle on the major axis (2a) as diameter. Thus the e:,11 under the Roulette (an Elliptic Catenary. lee 8) of a focus as the Ellipse rolls upon a linD is 2na2. XI. If Anz curve roll upon a line, the are length of the Roulette described by a 2sInt is equal to the corresponding arc length of the Pedal wlth respect to the generating point. For example The length, 8a, of one arch of the ordinary Cycloid is the same as that of the Cardioid, The length of one arch of the Elliptic Catenary is 2na, the circumference of the circle on Ve major axis of the Ellipse,

3. THE LOCUC OP THE CENTER OP CURVATURE OF A CURVE, MEASURED AT THE POINT OF CONTACT, AS THE CURVE ROLLS UPON A LINEt Let the rolling curve be given by its Whewell 18o ROULETTES intrinsic equation: s = f(y). Then, if' x,y are coordinates of

y the center of curvature, x = s = r(Y), Y = R = f'(i) are parametric equations of the locus. For example, for the

R Cycloidal family,

s A.sin B y x = A'sin Bp, y = AB'cos By and the locus is

B2x2 y2=A2B2 , anEllipse. Fig. 163

4. THE ENVELOPE OF A LINE CARRIED BY A CURVE ROLLING UPON A FIXED LINE: Draw PQ perpendicular to the carried line. Then Q is the point of tangency of the carried line with its envelope. For, Q has, at the instant pictured, the direction of the carried line and every point of that line has con- .- P\ ter of rotation at P. The envelope is thus the locus of points Q. Fig. 16 Let the curve roll to a neighboring point Pi carrying Q to Q1 throuch the angle dry. Then ifs represents the arc length of the envelope,

da Qy + TQA siny'ds z'dY,

stny(--,2)+ 2 dy dtf a relation connectirw radii orcurvature or rollin curve and envelope, tntrinsic equations or theenvelope are frequfAlt1:,' easily obtained. For example, considerthe thvi1(pr a diameter or a civole or vadlus a.Hero ROULETTES 181 z = a'sinp

and ds = a. dy Thus 0 = 2a.siny and dy

ra = -2a.cosd , an intrinsic equation of an ordinary Cycloid,

Fig. 16,

5. THE ENVELOPE OF A LINE CARRIED BY A CURVE ROLLING UPON A FIXED CURVE: If one curve rolls upon another, the envelope of a carried line is given by

= (cos a).----- dy (RI + R2) where tha normals to line and curves meet at the angle a, and the ills are radii of curvature of the curves at their point of contact.

Fig. 166 182 ROULETTES 6. A CURVE ROLLING UPON AN EQUAL CURVE: Its one curve rolls upon an equal fixed curve with corrb spondAng points in contact, thAe whole configuration is a refleo- tior in the common tangent (Maclaurin 1720). Thus the Roulette of any carried point 0 is a curve similar to the pedal with respect to 01 (the reflection of 0) with double its linear Oimenslons. A simple illustration Is the Cardioid. (See Caustics.)

Fig.167

7. :SOME ROULETTES;

Rolling Curve Fixed Curve Carried Element Roulette

Elliptic Cate. Ellipse Lino Focus narY4

Eyperbolic Cato Line FOCUA ByTerbola nary*

Recip:ocal Lino Polo Tractrix Spiral --, Involute of Line Center of Circle Parabola Circle

Cycloidal Line Center Ellipse Family Involute of the Line Any Curve Point of Line Curve

Curve similar to ;goal Curve Any Point Any Curve Pedal ROULETTES 183 SOME ROULETTES (Continued): 11

Rolling Curve Fixed Curve Carried Element Roulette

Equal Parabola Vertex Ordinary Cissoid Parabola

Circle Circle Any Point Cycloidal Family

Parabola Line Directrix Catenary

Involute of Circle Circle Any Line Epicyoloid

Involute of a Catenary Line Any Line Parabola

*The surfaces of revolution of these curves all have constant mean curvature. They appear in minimal problems (soap films).

8. The mechanical arrangement of four bars shown has an action equivalent to Roulettes. The bars, taken equal in pairs, form a crossed parallelogram. If a smaller side AS be fixed to the plane, Fig. 168(a), the longer bars intersect on an Ellipse with A and B as foci. The points C and D are foci of an equal Ellipse tangent to the fixed one at P, and the action is that of rolling Ellipses. (The crossed parallelogram is used as a "quick return" mechanism in machinery.)

Fig, 168 184 ROULETTES On the other hand, if a long bar BC be fixed tothe plane, Fie. 168(b), the short bars(extended) meet on an Hyperbola with B and C as foci. Upon thisHyperbola rolls an equal one with foci A ani D, theirpoint or contact at P. If P (the intersection of the longbars) be moved along a line and toothed wheels placed on thebars BC and AD as shown, Fig. 169(a), the Rouletteof C (or D)

r (

Fig. 169

Is an Elliptic Catenary, a planesection of the Undulpid whose mean curvature is constant.The wheels require the motion of C and D to be at right anglesto the bars in order that P be the center ofrotation of any point of CD. The action is that of anEllipse rolling upon the line. If the intersection of the shorterbars extended, Fig. 169(b), with wheels attached, movealong the line, the Roulette of D (or A) is theHyperbolic Catenary. Here A and b are foci oC theHyperbola which touches the line at P. ROULETTES 185

BIBLIOGRAPHY

Aoust; CJUV1;03 Planes, Paris (1873) 200. Besant, W. H.: Roulettes and Glissettes, London (1870). Cohn-Vossen: Anschaullehe Geometrie, Berlin (1932) 225. Encyclopaedia Britannica:,"Curves, Special", 14th Eth Maxwell, J. C.: Seientifig Papers, v 1 (1849). Moritz, R. E.: U. of Wash. Publ. (1923). Taylor, C.: Curves Formed 1).,ythe Action of ... Geometric Chucks, London (1874). Wieleitner, H.: Spezielle ebene Kurven, Leipsig (1908) 169 ff. Williamson, B.: Integral Calculus, Longmans, Green (1895) 20!) ff., 238. Yates, E.G.: Geometrical Tools, Educational Publishers, St. Louis (1949). SEMI-CUBIC PARABOLA

IISTORY: ay2 = x3was the first algebraic curverectified (Neil 1659). Leibnitz in 1687 proposed the problem of finding the curve down which a particle may descend under the force of gravity, falling equal verticaldistances In equal time intervals with initial velocitydifferent Crom zero. Huygens announced the solution as aSemi-Cubic Parabola with a vertical cusp tangent.

DESCRIPTION: The curve is defined by the equation:

y2 = Ax3 + Bx2 + Cx + D = A(x - a)(x2 + bx +c) , which, from a fancied resemblance tobotanical items, is sometimes called a Calyx and includes formsknown as Tulip, Hyacinth, Convolvulus, Pink, Fucia,Bulbus, etc., according to relative values of the constants.(See Loria.) In sketching the curve, it will be foundconvenient to draw as a vertical extension theCubic Parabola.(See Sketching, 10.) yi = y2. Values for which yl is negativecorrespond to imaginary values of y. There is symmetry withrespect to the x- axis. For example: SEMI-CUBIC PARABOLA 187 yl= = (x-1)(Y-2)(Y-A yl =y2 =(x-1)(x-2)2

Fit. 170

Slope at x = 1 (etc.): Slope at x = 2 (etc.): Limit y Limit y = 0(-1, x-, 21(x-2)

Limit\/(x-2)(x-.3) Limit + 077-r 1. X 4 1 X"1

(NOTE: Scales on X and Y-axes difi'erent).

2. GENERAL ITEMS: (a) The Semi-Cubic Parabola 27ay2= 4(x - 2a)3 is the Evolute of the Parabola y2 . 4ax. (b) The Evoiute of aye= x3 is

a'a - 18x)3 = (54ax + (*)y2 + a2)2 .

BIBLIOGRAPHY

Lorla, G.: Spezielle Algebraische and Transzendte ebene Kurven, Lelpsig (1902) 21. SKETCHING

ALGEBRAIC CURVES: f(x,y) = U.

1. INTERCEPTS - SYMMETRY - EXTENT are items to be noticed at once.

2. ADDITION OF ORDINATES: The point-wise construction of some functions, y(x), is often facilitated by the addition of component parts. For example (see also Fig. 181):

ytl

x 4 .11 .)1.ah /1 . . coo 1 c, 4, ./..z c1/4, 1 y,

1 t;, 5, :411t,/ I;

Fig. 171

The general equation of second degree:

Axe + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0 (1) may be discussed to advantage in the same manner. Rewriting (1) as

Cy = Bx- E (1177-AC)x2+ 2(BE- CD)x+E2- OF, C 0, we let Cy = yl yv, SKETCHING 189 where yl = -Bx - E, ( 2) and y2 = VA82 AC)x2 2(BE - CD)x + E2 - CF. ..(3) More Y22 (B2- AC)x2- 2(BE- CD)x - E2+ CF = 0, in which it is evident that the conic in (3) or (1) is an Ellipse if B2 - AC < 0, an Hyperbola if B2 - PC > 0, a Parabola if B2-AC =O. The construction is effected by combining ordinates in (2) and (3):

Fig. 172

Some facts are evident: (a) The center cf the conic (1) Is at CD - BE AE - BD

X B2 - AC ' e 132 - AC (b) Since yl = -Bx - E bisects all x = k, this CD - BE line is conjurate to the diameter x = - AC In the case the Parabola, yi = -Bx- E is paranel to the axis of symmetry. Thls axis of symmetry is thus i-B% inclined at Arc tan( --) to the x-axit. The point of

tanF,:ency of the tangent with slope is the vortex of the Parabola. 190 SKETCHING (c) Tangents at the points orIntersection of the line yl =-Has - E and the curve(1) are vertical. (In coralectIon, see Conics,).

of 3. AUXILIARY ANDDIRECTIONAL CURVES: The equations simpler and more some curves may beput into forms where familiar curves appear ashelpful guides in certain regions of the plane. For example:

2 77- y = e cosx -x )X

Fig. 173

Ir the neighborhood of the The quantity ex here con- 1 trols the maximum and or' .in --- dominates and the x mirimum values of y and is given curve follows the called the damping factor. 1 The curve thus oscillates Hyperbola y = - As 3x between y = ex and y = -ex since cosxvaries x oc, the term x2 domi- r nates and the curve follows only between -1 and +1. the Parabola y = x.

(See also Fig. 92.) SKETCHING 191 4. SLOPES AT THE INTERCEPTPOINTS AND TANGENTS AT THE ORIGIN: Let th.,, given curvepass through (a,0). A line through this 1.,:int and a neighboring point (x,y)has slope:

Limit y (x-a) x 4 a .(x-a) m s the slopc of the curve at (a,0).

Fig. 1714

For example:

y = 2x(x 2)(x- 1) Y2 = 2x(x- 2)(x - 1) Limit has m = has m =Limitimit y x 4 2 (x-2) x 2 (x-2)

Limit , , Limit 2xlx-1! = 4 i2x(x-Al x 2 x 4 2± (x-2) for its slope at (2,0).

for its slope at (2,0).

If a curve passes through the origin, itsequation has no constant term and appears:

0 = ax + by + cx3 + dxy + ey2 + fx3 +so,' or 0 = a + b4) + cx + dy + ey(Y)+ fx3 +

Taking the limit here as both x andy approach zero, the quantity approaches m, the slope of the tangent at (0,0):

0 = a + bm or m = .1--/ whence [ax 4. by.61.

Thus the collection of terms of first degree setplUal to zero, is the equation of the tangent at the origin. 192 SKETCHING If, however, there are no linear terms.the equation of the curve may he written:

0 c +d(1) + e(2)2 + fx +

and 0 = c + dm + em2 gives the slopes mat the origin.The tangents are,

netting m = y :

or 0 = cx2 + dxy 4. "2 I 0 . c +d(1) + e(1)2 '

It is now apparent that thecollection of terms of lowest degree set equaltorero is the equation of the tangents at the origin. Three casesarise (See Section 7 on SingularPoints): if this equation has no realfactors, the curve has no real tangents andthe origin is an isolated point of the curve; if there are distinct linearfactors, the curve has distinct tangents and the origin is anode, or multiple point, of the curve;

if there are equal linear factors,the origin is generally a cusp point of the curve.(See Illustrations, 9, t'or a:isolated point where a cusp isindicated.) For example: 2 x2(1-x) y2 = x2(x-1) = V2 = X3

Pig, 175

has (0,0) a3 a I has (0,0) as a has (0,0) as an I isolated point node cusp SKETCHING 193 5. ASYMPTOTES: FOP purposes ofcurve sketching, an avymptoto is defined as "a tangentto the curve at In- finity". Thus :s askod that the liney = mx + k meet the curve, generally, in two infinitepoints, obtained in thrs fashion of a tangent. Thatis, the simultaneous solut:on of

f(x,y) Tr. 0 and y = MX k n 1-1 or anx + an_Ixt n-2 + an-2Y + + alx + ao = 0..(1)

where the a's are functions ofm and k, must contain two roots x = =. Now if an equation

n aoz + alzn'*1 + + an.lz + an =0 (2) has two roots z = 0, then an = an.1 = 0. But if z = , ths equation reduces to the preceding.Accordingly, an equation such as (1) has two infinite rootsif = an_i = 0.

To determine asymptotes, then, set thesecoefficients equal to zero and soave for simultaneousvalues of m and k. For example, consider the Folium: x3 + y3 3xy = O. re sr:= mx k:

(1+m3)x3 3m(mk-i)x2

3k(mk-1)x k3= 0. For an asymptote:

1 + m3 = 0 or m -1 and 3m(mk-1) =0 or k -1. Thus x + y + 1 = 0 is the asymptote.

Fie. 176 _111Mmii,_

194 SKETCHING OBSERVATIONS: Let Ill, Qn be polynomialfunctions of x,y of the nth degree, each of whichintersects a line in n points, real or imaginary. Suppose agiven polynomial function can be put into the form:

(y - mx - a).P11.1 +-110 -1 = 0. (3) Now any line y = mx k cuts this curve once atinfinity since its simultaneous solutionwith the curve results in an equation of degree(n-1). This family of parallel lines will thus contain theasymptote. In the case of the Flium just given: (y + x)(x2 - xy + ya) - 3xy = 0,

the anticipated asymptote hasthe form: y + x - k = 0 and the value of k is readilydetermined.* Suppose the given curve of thenth degree can be written as: (Y - mx - k).Pn_i + Q11.2 = 0. (4)

Here any line y - mx - a =0 cuts the curve once at infinity; the line y - mx k = 0 in particular cuts __twice. Thus, generally, thislatter line is an asymptote, For example:

*Thus: 5- 3x), + x2 + yg y 2 3. - + (17)

here -1. As x,y- a, - - 1 and the last term 1 . ( .1) + 1 X

Thus y = -x - 1 is the Asymptote. SKETCHING 195

y3 - x3 + x o 1 (2y + x ) ( y x) -1 m 0

Fig, 177

has y = x for an asymptote. 1 has asymptotes 2y + x = 0, y - x a 0*

The line y = mx + k meets this curve (4) again in points which lie on Qn.2 m 0, a curve of degree (n-2). Thus the three possible asymptotes of a cubic meet the curve again in three finite points upon a line; the four asymptotes of a quartic meet the curve in eight further points upon a conic; etc. Thus equations of curves may be fabricated with specified asymptotes which will intersect the curve again in points upon specified curves. For example, a quartic with asymptotes x= 0, y 0, y - x = 0, y + x = 0 meeting the curve again in eight points on the Ellipse x2 + 2y2 se 1, is: xy (x2 - y2) - (x2 + 2y2 - 1) = 0.

* In fact, any conic whose equation oan be written as (y.ax)(7.bx)+on0 has asymptotes and is accordingly a Hyperbola, 196 SKETCHING 6. CRITICAL POINTS:

(a) Maximum-minimum valuesof occur at points (a,b) for which

. 0 m dx

with a change in sign of this derivativeas x passes through a.

Maximum- minimum values of x occur at those points (a,b) for which

dx dy O' m

with a change in sign of this derivativeas y passes through b. For example:

y2= x3(1 - x) y3= (x - 1)2(x + 1)9

"N. .r

Fig. 178

(b) A Flex occurs at the point (a,b) for which (if Y" is continuous)

y" = 0, Co with a change in sign of this derivative as x passes through a. For example, each of the curves: n

SKETCHING 197 3 Y = x3s y "0 = 0 y =x5, y "o = c°

Fig. 179

has a flex point at the origin. Such points mark a change in sign of the curvature (that is, the center of curvature moves from one side of the curve to an opposite side). (See Evolutes.) Note: Every cubic y = ax3 +bx2 + cx + d is symmetrical with respect to its flex.

7. SINGULAR POINTS: The nature of these points,when located at the origin, have already been discussed to some extent under (4). Care must be taken,however, against immature judgment based upon indications only. Properly defined, such points are those which satisfy the conditions:

assuming f(x,y) a polynomial, continuous and differenti- able. Their character is determined by the quantity:

F (fxy)2 fx afYY That is, for F < 0, an _isolated (hermit) point,

F = 0, a .P.01), F > 0, a node (double point, triple point,etc.). 198 SKETCHING

Thus, at such a point, the slope:Ai= -v.,;; has the LY 0 indeterminateform . 0 Variations in character are exhibited in the examples which follow (higher singularities, suchas a Double Cusp, Osculinflexion, etc., are compoundedfrom these simpler ones).

8. POLYNOMIALS: y = P(x) where P(x) isa polynomial (such curves are called "parabolic").These have the following properties:

(a) continuous for all valuesof x; (b) any line x= k cuts the curve in but one point;

(o) extends to infinity in twodirections; (d) there are no asymptotesor singularities;

(e) slope at (a,0) is Limit(V211]as x -4 a;

(f) if (x-a)k is a factor of P(x),the point (a,0) is ordinary if k = 1; max-min. if k is even;a flex if k is odd ( 1). BEST COPYAVAILABLE

9. ILLUSTRATIONS:

Fig. 180 MINIABLE BESI 200 SKETCHING ILLUSTRATIONS (Continued):

X I 1--) y Yn !FY' /

Fig. 181 SKETCHING 201 10. SEMI-POLYNOMINALS: y' = P(x) where P(x) is apoly- nomial (such curves are called"semi-parabolic"). In sketch 1141 semi-paraboll(! curves, it may be found ex- pedient to sketch the curve Y = P(x) and from this ob- tain the desired curve by taking the square root or the ordinates Y. Slopes at the intercepts should be checked as Indi.:ated In (4) The example s.lown Is

Y=y2 =X(5 - x)(x - 2)2. In projectin, the maximum Y's and v's occur at the same x's; negative Y's yield no correspondim y's; the slope at (2,0) is Limit y Limit X x 2 (x-2)=x 4 2 l).(3-71 = +v.

Fig. 182

11. EXAMPLES: (a) Semi-Poi momials x2 y2 y2= x(1 X2) y2 = x2( 1 - x)

y2= x2(x 1) =X2( 1 X3) y2=x3( 1- x) 3( y2 x4( x3 y2= X X 1) y- = x4(1 -X3)

=xi( 1-x2)y2-x4( 1 x4) y2 = x5(x 1)4 x2(x2 x2 4)3 y2= ( 1 -x`')3 y2= x( x 1)( x - 2) y2= 202 SKETCHING (b) Asymptotes:

Ft2 x2 112 (yr x2y +y2x = : [lc= 0, y= 0, x+ y= 0].

y3= x(Et2 x`) [x+ y =0]. x3 + y3= a3 [x + ym0]. 2a, x3- a(xy+ 0 :[ x=o] .(2a e)= -x)x2-y3=0 :[x+ y=-3 J.

y2( x2-y2)-21,y3+213x =0 : [y = 0, x y = a, x+ y+ 0].

AY-x)2( Y + 2x)=9913. (3' - b)(Y -c)x2=a2y2. 2 2 X2 x2y2 ay2+ b 0. ( x y) xy- a( x + y) =b3.

( x-y) 2(x-2y)(x-3y) - 2a( x3-y3)-2a2(x+y)(x-2y) =0 : [four asymptotes].

3, . 23 x2( x+y)( x-y) 2+ ax x-y) - ay = 0 :[x = t a, x-y+a= 0,x-y= , 2

x+y+ = 0], 2 (x2 y2)(y2 2y3 x2 _ -4X2-610 + 5x2y 3xy2 - x,y 1 = 0

has four asymptotes which cut the curve again in eight points

upon a circle.

4(x4 + y4) 17x2y2 4x(4y2-x2) + 2(x2- 2) =0 has asymptotes that cut the curve again in points upon the Ellipse

x2 45,2 + 4,

a(y-x)2 x3[Cusp]. (2y+x+1)2=4(1-x)5[ Cusp ],

(y-2)2= x(x-1)2[ Double a3y2 2abx2y =x5 [Osculin- Point] flexion] x4 2x2y xy2 y2 y2 2x2y x4y_ - + 0 [Cusp x4=0 [Double of second. kind at origin] cusp of second. kind at origin], y2 2x2y x4y y2= 2X2Y + X4Y 2x4 (Iso- xx4 rDoublt, lated Pt ], Cusp]. 3 2 2y 2 x+ 2x+ 2xy. y2 +5x . 2y X4- 2ax - axy +a2y2e0 = 0 [Cusp of first kind], [Cusp of second kind.], SKETCHING 203 12, SOME CURVES AND THEIR NAMES: Alysoid (Catenary if' e = e): aR . c2 + s2. Jowditch Curves (Lissajous): x = Asin(nt + c) y = b.sin t [ (See Osgood's Mechanics for figures).

a2 2 Bullet Nose Curve: m - = 1.

Cartesian Oval: The locus of points whose distances, rl, r2, to two fixed points satisfy the relation: rl + mr2 = a. The central Conics will be recognized as special cases. Catenary of Uniform Strength: The form of a hanging chain in which linear density is proportional to the tension.

n u Cochleold: r =a.(si8 --). This isa projection of a cylindrical Helix. Cochloid: Another ram-for the Conchoid of Nicomedes. y2). Cocked Hat: (x2 + lay - a2)2 =y2(8.2 2 a b Cross Curve: m + m . 1.

Devil Curve: y4 + aye - y4 + by2 . O. This curve is found useful in presenting the theory of Riemann surfaces and Abelian integrals (see AMM, v 34, p 199). Epi: r.cos 1(0 = a (an Inverse of the Roses; a Cotes' Spiral). Folium: The pedal of a Deltoid with respect to a point on a cusp tangent. Oerono's,Lemniscate: x4 = a2(x2 - y2). Hippoptilt of Eudoxus: The curve of intersection of a circular cylinder and a tangent sphere. Horopter: The intersection of a cylinder and aHyper- bolic Paraboloid, a curve discovered by Helmholtzin his studies of physical optics. ,11Hospital's Cubic: Identical with the Tschirnhausen Cubic and the Trisectrix of Catalan. 2O SKETCHING SOME CIMVES AND THT,R NAMES (Continued):

uc!exus: Et24 , b4(i ti:!ed by Eudoxu:; the pc,

Kappa Curve: y2 x2 F si2) = a2x2, orr = t acotO.

n Lamt: Curves: () + ("') . 1. (See Fvolutes).

Pearlf: c:r Shwe: y" k(a - x)P.xn, where the exponents are pe::ItIvc :nte,-ers. b2y2 = x3(a - x). Pear shaped. See this section 6(a). Poinsot's rcosh kU = a. 2aU REadvatrox of jipplas: r.sin U = --- n

Rh3duneae (Hose.) :r = a.-!os k U. Theve are Epitro- chotdn. Sem!-Tr:dent:

xy` = a3 : Palm Stems.

xy2 = Th2(1? - x) : Archer's Bow.

x(y2 + b') = aby : Twisted Bow.

x(:i2- b2) = ahy : Pilaster.

x(y2 - h2) ab2 : Turinel.

xy2 = m(x2 + 2bx + 1.12 c`): Urn, Goblet.

b2xy2 . (a - x)3 : PyramId.

(.22yy2 ,.-, (a - - Y)2 : Festoon, Hillock, Hel- met.

d2xy2 ( a)(x b)(x - c): owe P(..t., Trophy, Swir and Chair, Crane. Seuentino: A projectior, or the firopter.

Per-us: Syet:on a turns by plare:! taken varallel Its axls.

Svnta(!tr:x: ,r a pc:;:.nt'>r the lanr:ent to a Tract:x a% a Crom thy. Q.,' tan- g,!ncy. SKETCHING 205 SOME CURVES AND THEIR NAMES (Continued): Trident: xy = ax3 + bx2 + cx + d. Trisectrix of Catalan: identical with the Tschirn- hausen Cubic, and l'Hospital's Cubic. y2) a(y2 3x2). A Trisectrix of Maclaurin: :.(x2 + curve resembling the Folium of Descartes which Maclaurin used to trisect the angle.

Tschirnhausen's Cubic: a = r.cos3 , a Sinusoidal

Spiral. Versieraf Identical with the Witch of Agnesi. This is a projection of tbe Horopter. Vivianl's Curve: The spherical curve x a.sin c cosy, 2 y = wcos Cf, z = a.sin y, projections of which include the Hyperbola, Lemniscate, Strophoid, and Kappa Curve.

;ix=xy: See A.M.M.: 28 (3921) 141; 38 (1931) 444; oat. (1933).

BIBLIOGRAPHY Echols, W. H.: Calculus, Henry Holt (19n8) XV. Frost, P.: Curve Tracing, Macmillan(1892). Hilton, H.: Plane Algebraic Curves, Oxford(1932). Loria, G.: Spezielle Algebraische and Transzendente Vbene Kurven, Leipsig(1902). Wieleitner, H.: Spezielle &fine Kurven, Leipsig(1908). SPIRALS

HISTORY: The investigation of Spirals began at least with the ancient Greeks. The famous Equiangular Spiral was discovered by Descartes, its properties of self- reproduction by James (Jacob) Bernoulli (1654-1705) who requested that the curve be engraved upon his tomb with the phrase "Eadem mutate resurgo" ("I shall arise the same, though changed").*

1. EQUIANGULAR SPIRAL: 1r=a.ee'°"41.(Also called Logarithmic from an equivalent form of its equation.) Discovered by Descartes in 1638 in a study of dynamics.

rig, 183

(a) The curve cuts all radii vectores at a constant

angle 4. ("Ifr m tan a).

Lieumen, W.: Luetigee and Merkwurdigee von Zehlen and Porten, p. 40, gives a picture or the tombstone. SPIRALS 207 dr (h) Curvature: Since p = v.sin a, R = r.csc a =CP dp (the polar normal). R = s.cot a.

dr (Sk)r.g.2% sin a, (c) Arc Length: Toi ) = (r.cot `1:10"ds and thus s = r.sec a = PT, where s is measured from the point where r = C. Thus, the arc length is equal to the polar _tangent (Descartes), (d) Its pedal and thus all successive pedals with respect to the pole are equal Equiangular Spirals, (e) Evolute: PC is tangent to the evolute at C and angle PCO = a. OC is the radius vector of C. Thus the first and all successive evolutes are equal Equi- angular Spirals. (f) Its inverse with respect to the pole is anEqui- angular Spiral. (g) It is, Fig. 184, the stereographic projection (x = k tan cos 0, 2 y = k tani.sin 0) of a Loxodrome (the curve ;:utting all meridians at a constant angle: the course of a ship holding a fixed direction on the compass), from one of its poles onto the equator (Hal- ley 1696).

(h) Its Catacaustic and Diacauatic with the light source at the pole are Equiangular, ,Spirals. (i) ;Aengths of radii drawn at equal angles to each other f011m a geometric progression. (j) Rout If the spiral be rolled along a line, the path of the pole, or of the center of curvature of the point of contactvls a straight line. COP1 MIAMI BEST

208 SPDXALS (10 The septa of the Nautilus are Equiangular Spirals. The go. curve seems also to appear in the arrangement of seeds in the sunflower, the forma- tion of pine cones, and other growths.

4 \

Fig. 1B5

(1) The limit of a succession of Involutes ofan given curve is an Equiangular Spiral. Let the given curve be a = f(e) and denote by 6n the arc length of an nth involute. Then all first involutes are given by

ci + Odo = co + r(o)do, 0 where c represents the distance measured along the tangent to the given curve. Selecting a particular value for c for all successive involutes

s2 =!tc+ ce + f(0)dej dO o

snto tO+002/21+ 03/31+ .;.+ f f(e)dO]ntha c SPIRALS 209 where this nth iterated integral may be shown to approach zero. (See Byerly.) Accordingly,

ori 02 03 = Limit sr), = e(e 4.-27+-57 ...... ) n =

or Is=c(ef/- p an Equiangular Spiral. (m) It is the development of a Conical Helix (See Spiral of Archimedes.)

2. THE SPIRALS: r = a0 Include as special cases the aaei Archimedean (due to following: 1r Conan but studied particularly by Archimedes in a tract still extant. He probably used it to gquare the circle). (a) Its polar subnormal is constant.

(b) Arc length from 0 to

9: e 14rF 3.034472-6

(Archimedes). r3 (c) A = -6--a- (from 0 = 0 118. 1136 to 0 r/a). (d) It is the Pedal of the Involute of a Circle with respect to its center. This suggests the description by a carpenter's square rolling without 04pping upon a circle, Fig. 187(a). Here uT = AB = a. LetA start at A', B at 0. Then AT = arc A'T = r w se. Thus B describes the Spiral of Archimedes while A traces an Involute of the Circle. Note that the center of rotation is T. Thus TA and TB, respectively, are normals to the paths of A and B. 210 SPIRALS

Fig. 187

(e) Since r = a0 and ;. = /41, this spiral has found wide use as a cam, Fig. 187(b) to produce uniform linear motion. The cam is pivoted at the pole and rotated with constant angular velocity. The piston, kept in contact with a sprin device, has uniform reciprocating motion.

(f) It is the Inverse of a Reciprocal Spiral with respect to the Pole.

(g) "The casings of centrifugal pumps, such as the German supercharger, folic this spiral to allow air which Increases uniformly in volume with each degree of rotation of the fan blades to be conducted to the outlet without creating back-pressure." - P. S. Jones, 18th Yearbook, N.C.T.M. (1945) 219. AVAILABLE BESTCOPY

SPIRALS 211 (ii) The crtho-

t. Ion

of a Conical Helix on a plane per- pendlcular tz: its axis is a Spiral or Archimedes. The development or this Helix, however, !s an

Equiana,,ular Spiral

188) .

188

1 Reciprocal (Varignon 1704). (Some- times called Hyperbolic because of its analogy to the equation xy = a). (a) Its polar sub-

tar-went is con-

stant. (h, Its asymptote as a units rrom the InItlal line. Limit risinU= U 0 Limit a.sin = a. Fie, 189 0 a 0 212 SPIRALS (c) Arc Lengths of all circles (centers at the pole) measured from the curve to the axis are constant (= a).

(d) The area wounded by the curve and two radii is proportional to the difference of these radii. (e) It is the inverse with respect to the pole of an Archimec'ean Spiral. (f) Roulette: As thr curve rolls upon a line, the pole describes a Tractrix. (g) It is a path of a Particle under a central force which varies as the cube of the distance. (See Lemniscate 4h and Spirals 31.)

rrT7.---)74 1r2=a201 Parabolic (because of its analogy to y2 = a2x) (Fermat 1636).

(a) It is the inverse with respect to the pole of a

Lituus.

Fig. 190

n = -1/2 r2 a2 Li taus (Cotes, 1722,. (Similar in form to an ancient Roman Gmpet.)

(a) The areas of all circular sector:; are constant 2

2 2 " AVAILABLE BESTCOPY

SPIRALS 213 (b) It is the inverse with respect to thc. pole of a Parabolic Spiral. (c) Its asymptote is the initial line.

Limit rsin 0= . 0 Fig. 191 Limit Nib sin0 - o. o

1 (d) The Ionic . _

diioi Volute: Together with other spirals, the Lituus is used as a volute in architectural design. In practice, the Whorl is made Fig. 192 with the curve emanating from a circle drawn about the pole.

3. THE SINUSOIDAL SPIRALS: rn = ancos n0 or rn = ansin nO. (n a rational number). Studied by Mac- laurin in 1718. (a) Pedal Equation: rn+1 = anp.

n (b) Radius ofCurvature: a r R = (n + 1)rn-1 ITTTY5 which affordsa simple geometrical method of con- structing thecenter of curvature. (c) Its Isoptic is another Sinusoidal Spiral. 214 SPIRALS

1 (d) It Is rectifiable ifis an integer.

(e) All positive and er;ative Pedalsare again Sinusoidal Spirals. (f) A body acted upon by a central force inversely proportional to the (2n + 3) power of its distance moves upon a Sinusoidal Spiral. (g) Special Cases: n Curve

-2 Rectangular Hyperbola -1 Line -1/2 Parabola -1/3 Tschlrnhausen Cubic 1/3 Cayley's Sextic 1/2 Cardioid

1 Circle

2 Lemniscate

(In connection with this family see also Pedal Equa- tions 6 and Pedal Curves 3). (h) Tangent Construction: Since rfl-1 pl - ansin nO,

= - cot nO = cot(n - nO) = tan y

and y = nO - 2 which affords an immediate construction of an arbitrary tangent. SPIRALS 215 4. EULER'S SPIRAL: (Also called Clothoid or Cornu's Spiral). Studied by Euler in 1781 in connection with an investigation of an elastic spring. Definition: [ati dx =sin t.dt Va dy = acos t.dt, or [R.s=a2

2 2 S = 2at

Asymptotic Points:

a xo,Yo = 2 Fig. 193

(a) It is involved in certain problems in the diffraction of light. (b) It has been advocated as a transition curve for railways. (Since arc length is proportional to curvature. See AMM.)

5. COTES' SPIRALS: These are the paths of a particle subject to a central force proportional to the cube of the distance. The five varieties are included in the equation:

1 = + B. P

They are:

Fig. 194 I

216 SPIRALS

1. B = 0: the Equiangular Spiral; 2. A = 1: the Reciprocal Spiral;

1 3. = a.sinh n0;

1 4. = acosh n0;

i 5. = asin nO (the inverse of the Roses).

The figure is that of the Spiral r.sin 40 = a and its inverse Rose. The Glissstte traced out by the focus of a Parabola sliding between two perpendicular lines is the Cotes' Spiral: r.sin 20 = a.

BIBLIOGRAPHY

American Mathematical Monthly: v 25, pp. 276-282. Byerly, W. E.: Calculus, Ginn (1889) 133. Edwards, J.: Calculus, Macmillan (1892) 329, etc. Encyclopaedia Britannica: 14th Ed., under "Curves, Special." Wieleitner, H.: Spezielle ebene Kurven, Leipsig (1908) 247, etc. Willson, F. N.: Graphics, Graphics Press (1909) 65 ff. STROPHOID

HISTORY: First conceived by Barrow (Newton's teacher) about 1670.

1. DESCRIPTION: Given the curve f(x,y) = 0 and the fixed points 0 am A. Let K be the intersection with the curve of a variable line through O. The locus of the points PI and P2 on OK such that KP1 = KP2 = KA is the general Strophoid.

Fig. 195

2. SPECIAL CASES: If the curve f = 0 be the line AB and 0 be taken on the perpendicular OA = a to AB, the curve is the more familiar Right Strophoid shown in Fig. 196(a).

k.) -

Fig. 196

This curve may also be generated an in Fig. 196(b). Here a circle of fixed radius a rolls upon the line M (the 218 ST RO PROW asymptote) touching it at R. The line AR through the fixed point A, distant a units from M, meets the circle in P. The locus of P is the litht Strophold. For,

(oV)(VB) (VP)` and thus BP is perpendicular to OP. Accordingly, angle KPA = angle KAP, and oo

KP KA, the situation of Fig. 196(a). The special oblique Strophoid (Fig. 197(b)) is generated if CA is not perpendicular to AB.

Fig. 197 (b)

. This F:trophoid, formed when r = 0 is a line, can be identified as a Cissoid of a line and a cIrcle. Thus, in Fig. 197, draw the fixed circle through A with center at O. Let E and D be the Intersections of AP extended with the line L and the fixed circle. Then in Fig. 197(a): ED . a.cos 2ye fiec y and AP . 2a.tan 0.sin y = 2acot 2y,sin y.

ThusAP ED, and the locus of P, then, is the Cissotd of the line L and the fixed circle. STROPHOID 219 30EQUATIONS: Fir. 196(a), 191(a):

r = a(sec u an u),(Pole at o); or da - x Fig. 196(b) :

2 x2(a x) r = a(sec u- 2cos U),(Pole at A); or y = a x Ft 197(b):

r = a(sin a - sin u).csc(a U),(Pole at o).

4. METRICAL PROPERTIES:

A (loop, Fig. 196(a)) = a2(2-

5. GENERAL ITEMS:

(a) It is the Pedal of a Parabola with respectto any point of its Directrix.

(b) It is the inverse ofa Rectangular Hyperbola with respect to a vertex. (See Inversion). (c) It is a special Kieroiu. (d) It is a stero- graphic projection of Viviani's Curve. A (e) The Carpenter's Square moves, as in the generation of the Cis- soid (see Cissoid 4c), with one edm passing /r through the fLxed point S (Fig. 198) while its corner A moves along the line

Fig. 198 220 STEM PH= AC. If BC = AQ = a and C be taken as the pole of coordinates, AB = assecO. Thus, the path of Q is the Strophoid:

r = assec0 - 2a.cos0

BIBLIOGRAPHY

Encyclopaedia Britannica, 14th Ed., under "Curves, Special." Niewenglowski, B.: Cours de Geometrie Analytique, Paris (1895) II, 117. Wieloitner, H.: Spezielle ebene Kurven, Leipsig (1908). TRACTRIX

HISTORY: Studied by Huygens in 1692 and later by Leibnitz, Jean Bernoulli, Liouville, and Seltrami. Also called Tractory and Equltangential Curve.

Fig. 199

1, DESCRIPTION: It is the path of a particle P pulled by an inextensible string whose end A moves along a line. The general Tractrix is produced if A moves along any specified curve. This is the track of a toy wagon pulled along by a child; the track of the back wheel of a bicycle. Let the particle P: (x,y) be pulled with the string AP an a by moving A along the x-axis. Then, since the direction of P is always toward A,

y a2 222 TRACTPIX 2. EQUATIONS:

x = a.arc soonZ a2 y2 . a

x . a.ln(sec o + tan (i) - a.sin 0 y = aar-)50 [ s = a.ln see a2 + R2=a2e2241

3. METRICAL PROPERTIES:

(a) K = R = a.cotT

(b) A na2 [A 4 ftVa2-y2 dy (from par. 2, above . area of the circle shown)j. 2 (c) Vx 2.L1A (Vx = halfthevolume of the sphere of 3 radius a). (d) Zx = kna2 (2x = area of the sphere of radius a).

4. GENERAL ITEMS: (a) The Tractrix is an involute of the Catenary (see Fig. 199). (b) To construct the tangent, draw the circle with radius a, center at P, cutting the asymptote at A. The tangent is AP. (a) Its Radial is a Kappa curve. (d) Roulette: It 1.8 the locus of the pole of a Reciprocal Spiral rolling upon a etraight line. (e) Schiele's Pivot: The solution of the problem of the proper form of a pivot revolving in a step where the wear is to be evenly distributed over the face of the bearing is an arc of the Tractrix. (See Miller and Lilly.) RIfiailLE BESTCO

TRACT= 223

Fig. 200

(f) The Tractrix is utilized in details of mapping. (See Leslie, Craig.) (g) The mean or Gauss curvature of the surface generated by revolving the curve about its asymptote (the arithmetic mean of maximum and minimum curvature at a point of the surface) is a negative constant (-1/a). It is for this reason, together with items (c) and (d) Par. 3, that the surface is called the "pseudo-sphere". It Corms a useful model in the study of geometry. (See Wolfe, Eisenhart,Graustein.) (h) Prom the primary definition (see figure), it is an orthogonal trajectory of afamily of circles of constant radius wit. centers on a line, 224 TRAM=

BIBLIOGRAPHY

Craig: Treatise on Projections. Edwards, J.: Calculus, Macmillan (1892) 357. Eisenhart, L. P.: Differential Geometry, Ginn (1909). Encyclopaedia Britannica: 14th Ed. under "Curves, Special." Graustein, W. C.: Differential Geometry, Macmillan (1935). Leslie: Geometrical Analysis (1821). Miller and Lilly: Mechanics, D. C. Heath (1915) 285. Salmon, G.: Higher Plane Curves, Dublin (1879) 289. Wolfe, H. E.: Non Euclidean Geometry, Dryden (1945). TRIGONOMETRIC FUNCTIONS

HISTORY: Trigonometry seems to have been developed, with certain traces of Indian influence, first by the Arabs about 800 as an aid to the solution of astronomical problems. From them the knowledge probably passed to the Greeks. Johann Willer (c.1464) wrote the first treatise: De triangulis omnimodis; this was followed closely by others.

1. DESCRIPTION:

y s.n x y x:Se

Fig. 2C1

2. INTERRELATIONS:

(a) From the figure: (A + B + C = n)

a b c = = 2R sinA sinti = sinC

Lsin Aoll sin(B+0) = sinBcosC + cosBsinC cos(B+C) at cosBcosC - sinBsinC 226 TRIGONOMETRIC FUNCTIONS

(b) The Euler form:

Ix Z = e = cosx + isinx;

-ix = e =cosx- isinx;

%k (con + i.sinx) =

cos kx + i.sin kx

produces, on identify-

ing reals and

imaginaries:

Fig. 202

sin 2x = 2sinxcox cos 2x = 2cos2x - 1

sin 3x = 3sinx - 4siex cos 3x = 4c0s3x - 3cosx

sin 4x = 4sin cosx - 8siexcosx cos 4x = 800s4x - 8cos2x + 1 etc.

cos kx = 2cos(k-1)xcosx -cos(k-2)x sin kx = 2sin(k-1)xcosx - sin(k-2)x

(d) Since zk = cos kx + isin kx; Tk = cos kx -i.sinkx, k --k k k z + z = 2icos kx and z -z = 2isin kx. Thus to convert from a power of the sine or cosine into multiple angles, write

,z+-2 %n k k cosnx =k---), expand andreplace z + z by 2'cos kx 2 z_7%n k k sinnx= (---i , expand and replace z- z by 21sin kx, 2i with iE = 1. TRIGONOMETRIC FUNCTIONS 227 For example:

(1 - cos 2x) 2 (1 + cos 2x) sin2x= COSX - 2

(3nln x - sin 3x) 3 (cos3x + .5COS x) sin3x= COSX - 4 4

(cos 4x- 4cos 2x + 3) 4 (cos 4x + 4cos 2x + 3) sin4x COSX - 8 8

(sin 5x-5sin 3x+10sin x) (cos 5x+5cos 3x+10cos x) sinsx- COS X = 16 16

n +1 nx (e) sin x'sin 2 2 E sin kx = k=1 sin 2

n+1 nx cos x.sln 2 2 2; cos kx = k=1 in 2

(f) From the Euler form given in (b):

sin x = (ix), con x = cosh (ix) sin (ix) = i'sinh x, cos (ix) = cosh x

3. SERIES:

02 2k+1 2 (a)sin x ( -1) k x < c0 (2k+1): ) 0

2k x 2 on x = 2(-1)k x < (21) : 0 3 2 2 17 7 62 2/ tan x = x + + x- + x A 15 315 28 35 4 3 s 7 1 X X 2x x . 2 cot x = + .6., x < 12 p X 3 45 945 4725 = 1 2x = + x2 0112 k=1 228 TRIGONOMETRIC FUNCTIONS 2 x2 4 , 277 _8 2 n x 5x ol xe x+ ..., x <-+ 7 + 24 720 +8o64 .4*.

1 x 7 3 31 5 2 2 CSC x =-+ + + X < n X -6 360X 1120x+ "" = 1 2x .-+ 2 (-1)k x x2 - 2 k=1 1 x3 1,3 x5 135 x7 2 1, (b) ESTO Sir= = X + 0 4-. + --.. -- + 2 3 2.4 5 2.4.6 7 + " " 3c<

arc coex = - arc sin x. 2 3 S X X arcto:-1 x + ...,X2 <

1 1 1 n 1 x 1. 2 x 37 37 4' 57$$" arc cotx =- - arc tan x. 2

arc secx = - arcCSC X. 2 1 1 1 13 1 135 1 arc °SOX -+ x2 > 1. x 2 3x---24 -57g 246 73c7 + i.DIFFERENTIALS AND INTEGRALS:

dx d(sin x) = cos x dx d( arc sin x) 1 - x= -d( arc cos x) d( cos x) = -sin x dx dx d( arc tan x)= x) d( tan x) = sec2x dx 1 +x2d'arc cot (tract x) = -csc2x dx dx d(sec x) = sec x tan x dx d( arc sec x) - -d( arc csc x). xRr-T. d( csc x) -csc x cot x dx.

ftan x dx = In I sec xl

fcot x dx = In lain xI fsec x dx = In leec x + tan x x I ln I tan - fcac x dx = InIcac x- cot x 2 TRIGONOMETRICFUNCTIONS 229 5. GENERAL ITEMS: (a) Periodicity: All trigonometric functions are periodic. For example: 2n y = Asin Bxhas period: andamplitude: A.

has period: y = Atan Bx B (b) Harmonic Motion is defined by the differential equation:

+ B2.s=.(] .

Its solution is y = locos (Bt + in which the arbitrary constants are

A: the amplitude of the vibration,

T: the phase-lag.

(c) The Sine (or Cosine) curve is the orthogonal 2s2- jection of a cylindrical HeliL, Fig.203(a), (a curve cutting all elements of the cylinder atthe same angle) onto a plane parallel to the axis of the cylinder (See Cycloid 5e.)

(a) Fig. 203 ( b)

(d) The Sine (or Cosine) curve is theAevelopment of an Elliptical sectionof a right circular cylinder, Pig 203(b). Let the intersectingplane be 230 TRIGONOMETRIC FUNCTIONS

1 2 k and the cylinder: (z-1)2 x2 = 1 which rolls upon the XY plane carrying the point P :(x,y,z) into P1:(x=0,y). From the planes

y k(1-f).

But z = 1 - cos 0 1 - cos x.

Thus y (2)(1+ cosx)

A worthwhile model of this may be fashioned from a roll of paper. When slicing through the roll, do not flatten it. (e) A ma2 of a Great Circle Route:* Ifan airplane travelt on a great circle around the earth, the plane of the great circle cuts an arbitrary cylinder circumscribing the earth in an Ellipse. If the cylinder be cut and laid flat ai in (d) above, the .'round -theworld' course is one period of a sine curve. (f) Wave Theory: Trigo- nometric functions are fundamental in the development of wave theory. Harmonic analysis seeks to decompose a resultant form of vibration into the simple fundamental motions characterized by the Sine or Cosine curve. Fig. 204 This is exhibited in Pig. 205.

*This in nondoonformal (i.e.) angles are not preserved). TRIGONOMETRIC FUNCTIONS 231

sinxi.stn2x

Composition of Sounds. A tuning fork with octave overtone would resemble the heavy curve.

JJI

Four Tuning Forks in UnisonDoMiSolDo in ratios 4 : 5 t 6 t S.

Violin.

French Horn.

Pig. 205

(From Harkin's Fundamental Mathematics. Courtesy of Prentice-Hall.) 232 TRIGONOMETRIC FUNCTIONS Fourier Development.of a given function i. the composition of fundamental Sine waves of increasing fre- quency to form successive approximations to the vibration, For example, the "step" function y = 0, for -n

Y = n, for0

Pig. 206

the first four approximations of which are shown in Fig. 206.

BIBLIOGRAPHY

Byerly, W. E.: Fourier Series, Ginn (1893). Dwight, H. B.: Tables, Macmillan (1934). TROCHOIDS

HISTORY: Special Trochoids werefirst conceived by DUrer in 1525 and by Roemer in1674, the latter in connection with his study of the best formfor gear teeth,

1. DESCRIPTION: Trochoids areRoulettes - the locus of a point rigidly attached to a curve that rolls upona fixed curve. The name, however, is almost uni- AP. k versally applied to Epi- and Hypotrochoids(the path of a point rigidly //". attached to a circle rolling upon a fixed circle) to which the discussion here is restricted. (\

Fig, 207

2. EQUATIONS:

Epitrochoids Hypotrochoids

x m mcos t -kcos(mt/b) x = ncos t +kcos(nt/b)

y 0 msin t -lsin(mt/b) y m nsin t -ksin(nt/b) [ where m = a + b. where n m a - b, (these include the Epi- and Hypocycloidsif k - b). 234 TROCHOIDS 3. GENERAL ITEMS:

(a) The Limacon is the Epitrochoid where a= b. (b) The Prolate and Curtate Cycloids are Trochoids of a Circle on a line (Fig. 208):

Fig. 208

Consider generation by the point P [Fig. 209(a)) . Draw OP to X. Then, since arc TP equals arc TX, P was originally at X and P thus lies always on the line OX. Likewise, the diametrically opposite point Q lies always on OY, the line perpendicular to OX. Every point of the rolling circle accordingly describes a diameter of the fixed circle. The action here then is equivalent to that of a rod sliding With its ends upon two perpendicular lines - that is, a Trammel of Archi- medes. Any point F of the rod describe:, an Ellipse whose axes are OX and OY. Furthermore, any point G, rigidly connected with the rolling circle, describes an Ellipse with the lines traced by the extremities of the diameter through 0 as axes (Nasir, about 1290). Note that the diameter IS envelopes an Astroid with OX and OY as axes. This Astroid is also the envelope of the Ellipses formed 117 various fixed points F of Eg. (See Envelopes.) TROCHOIDS 235

leTrr-----T------,o!r' eG F

a 0

(a) Fig. 209 (h)

(d) The Double Generation Theorem (see Epicycloids) applies here. If the smaller circle be fixed[Fig. 209(b)) and the larger one roll upon it, any diamete:, RX passes always through a fixed point P onthe smaller circle. Consider any selected point S ofthis diameter. Since SO is a constant length and SO extended passes through a fixed point P, the locus ofS is a Limacon (see Limacon for a mechanismbased upon this). Accordingly, any point rigidly attached to the rolling circle describes a Limacon. If R be taken on the rolling circle, its path is aCardioid with cusp at P. Envelope Roulette: Any line rigidly attached to the rolling circle envelopes a Circle.(See Limacon 3k; Roulettes 4; Glissettes5.)

(e) The Rose Curves: r = a cos neandr = a sin nO are Hypotrochoids generated by acircle of radius - 1)arolling within a fixed circle of radius

na --- the generating point of the rollingcircle (n + 1) ' being ! units distant from its center.(First noticed by Suardi in 1752 and then byRidolphi in 1844. See Loris.) 236 TROCHOIDS

in) Fig. 210 ( b)

As shown in Fig. 230(b): OB = a, AB = b, OA = AP 2b act 121 bp, P . 2(a + u) m or 4 u, a- 2b Thus in polar coordinates with the initial line through the center of the fixed circle and a maximum point of the curve, the path of P is

a r = 2(a - b) cos(m + 0) = 2(a - b) cos a -2bU.

BIBLIOGRAPHY

Atwood and Pengelly: Theoretical Naval Architecture (for connection with study of ocean waves). Edwards, J.: Calculus, Macmillan (1892) 343 ff. Loria, G.: Apezielle algebraische and Transzendente ebene Kurven, Leipsig (1902) II 109. Salmon, G.: Higher Plane Curves, Dublin (1879) VII. Williamson, B.: Calculus, Longmans, Green (1895) 348 ff. WITCH OF AGNES!

HISTORY: In 1748, studied and named* by Maria Gaetana Agnesi (a versatile woman - distinguished as a linguist, philosopher, and somnambulist), appointed professor of Mathematics at Bologna by Pope Benedict XIV. Treated earlier (before 1666) by Fermat and in 1703 by Grandi. Also called the Versiera.

Fig. 211

1. DESCRIPTION: A secant OA through a selected point 0 on the fixed circle cuts the circle in Q. QP is drawn perpendicular 1,c, the diameter OK, AP parallel to it. The path of P is the Witch.

* Apparent4 the result of a misinterpretation. It seamy Agnesi.00n- fused the old Italian word "versorion (the name given the curve by Grandi) which means 'free to move in any direction' with 'versiera' vnioh means 'goblin', bugaboo', 'Devil's wife', etc. [See Scripts Mathematics, VI (1939) 2116 VIII (1941) 135 and School Science and Mathematics SINI (1940 57.] 238 WITCH OF AGNESI 2. EQUATIONS:

[x = 2a.tan u y(x2 + 4a2)= 8a3. Y = 2a.cos2U

3. METRICAL PROPERTIES:

(a) Area between the Witch and its asymptoteis four times the area of the given fixed circle (4na2).

(b) Centroid of this area lies at (0,11).

(e) Vi = 4n2a2.

(d) Flex points occur at U= f .

4. GENERAL ITEMS: A curve called the Pseudo-Witch is produced by doubling the ordinates of the Witch. This curve was studied by J. Gregory in 1658 and used by Leibnitz in 1674 in deriving the famous expression:

1 1 1 4 = 1 - + - + .

BIBLIOGRAPHY

Edwards, J.: Calculus, Macmillan (1892) 355. Encyclopaedia Britannica: 14th Ed., under "Curves, Special." INDEX

(Numbers refer to pages)

Addition of ordinates: 188 Cardioid: 4-7;17,61,84,91,124, Aeroplane design: 48 125,126,140,149,151,152,153, Agnesi, Marta Gaetana: 237 16,163,164,168,17:3,179,182, Alexander the Great: 36 214,235 Alysold: 203 Carpenter's Square: 28,50,209, Apollonius: 23,36,86,127,133 219 Arbelos: 25 Cartesian Oval: 149,203 Archer's Bow: 204 Cassinian Curves: 8-11;143,144 Archimedes Spiral: (see Catacaustic: (see Caustics) SpIrals, Archlmedean) Catalan's Trisectrix: 203,205 Archimedes Trammel: 3,77,108, Catenary: 12-14;20,63,80,87,117, 120,234 124,126,174,177,182,183,203, Astroid: 1-3;63,73,78,84,109, 222 111,126,140,156,157,163,169, Catenary, Elliptic: 179,182,184 174,234 Catenary, Hn.rbolic: 182,184 Asymptotes: 27,87,129,193,194, Catenary of Uniform Strength: 195,202,211,213 174,203 Auxiliary Curves: 190 Caustics: 15- 20;5,69.71,73,79, E,1,87,149,152,153,160,163, Barrow: 217 207 Bellavitis: 127 Cayley: 751 Sextic: 87,153,163, Beltrami: 221 214 Bernoulli: 1,12,65,67,68,69, Central force: 8,145,212,214, 81,93,143,152,175,206,221 215 Besant: 108,175 Centrifugal pumps: 210 Besicovitch: 72 Centrode: 119 Boltzmann: 106 Cesttro: 123,125,126 Bouguer: 170 Chasles: 85,119,138 Bowditch Curves: 203 Circle: 21-25;1)5,16,17,20,30) Brachistochrone: 68 31,61,69,79,91,126,127,128, Brianchon: 48 135,138,139,140,149,162,165, Brocard: 171 168,171,172,174,180,182,183, Balbus: 186 214,r03,233,'35 Bullet Nose Curve: 203 Cissoid: 26.30;20,126,129,141) 142,143,161,163,183,218,219 Calculus of Variations: 80 Clairaut: 77 Calyx: 186 Clothoid: (see Spirals) Euler) Cams: 6,137,210 Coohleoid: 203 24o INDEX

Cochloid: 203 Pe Mo!vre: Cocked Hat: 203 Decargues: 17' Compasv Constructicn: 1°8 !)0f.-tr'es: 6506,205,206,207 Conchoid: 31-33;30,108,109,1'0, Devi c.ion, standard: 96 121,141,142,148,203 Devil Curve: 203 Cones: 34-35;37,36,39 Diacaustio (see Caustics) Conics: 36-55;20,78,79,87,88, Diffraction of light: 215 112,130,131,138,140,149,156, Differential equation: 75,77 163,173,188,189,195,203 Diocles: 26,129 Convolvulus: 186 Directional Curves: 190 Coronas: 81 Discontinuous Curves: 100-107 Cornu's Spiral: (see Spirals, Discriminant: 39,57,76,189 Euler) Double generation: 81 Cotes: 212; Spiral (see Spirals, Duality: 48 Cotes') DUrer: 175,233 Crane: 204 Critical Points: 196 "e": 93,94 Crone Curve: 203 Elastic spring: 215 Crossed Parallelogram: 6,131, Ellipse: 36-55;2,19,27,63,78,79, 158,183 88,1u9,111,112,120,139,140, Cube root problem: 26,31,36,:04 149,157,158,164,169,173,178, Cubic, l'ilospitalls: 203,P0') 179,180,182,183,184,189,195, Cubic Parabola: 56-59:89,186,19.' '02,299,230,234 Cubic, Tschirnhausen: vlliptic Catenary: 179,182,184 214 rnvelopes: 75-80;2,3,15,50,72, Curtate Cycloid: 65,69 (see 1'3,:k5,87,91,108,109,110,111, Trochoids) 112,135,139,144,153,155,160, Curvature: 60.64;53,36,167,172, 161,175,180,181,234,235 180,181,184,197,207,213,215, Fpi: 203 223 Epicycloid: 81.85;4,5,63,87,122, Curvature, Constructif,n of Cen- 126,139,152,163,169,174,177, ter of: 54,55,145,150,213 180,182,183 (Mop: 20,27,90,192,197,199,200, Epitrochoids: (see Trochoids) 202 Equati'n of second degree: 39, Cylinder: 229,230 188 Cycloid: 65-70;1,4,63,80,89,92, Equiangular Spiral (see Spirals, 122,125,126,136,137,138,139, Equiangular) 172,174,176,177,179,180,181, Equitangential Curve: (see 182,183, (see also Epicycloids) Tractrix) Eudoxus, Hippopede of: 203 Damping factor: 190 Eampyle of: 174,204 da Vinci, Leonardo: 170 Euler: 67,71,82 Deltoid: 71.44; 84,126,140,164, Euler form: 94,116,226 169,174,203 Euler Spiral (see Spirals, Eu1er) INDEX 241

Evolutes: 86- 92;2,5,15,16,19,20, Hathaway: 171 57,66,68,72,79,85,135,139,149, Helix: 69,203,209,211,229 152,153,155,173,187,197,204, Helmet: 204 207 Helmholtz: 203 Exponential Curves: 93-97;20 Hessian: 99 Hillock: 204 Fermat: 237 Hippias, Quadratrix of: 204 Fermat's Spiral: (Bee Spirals, Hippopede of Eudoxus: 203 Parabolic) Hire: 138,175 Festoon: 204 Horopter: 203,20.205 Flex point: 10,56,87,90,196,198 l'Hospital: 68; c.aic: 203,205 Flower Pot: 204 Huygens: 15,66,67,86,135,152, Folium of Descartes: 98-99;193, 155,186,221 205 Hyacinth: 186 Foil= 72; (Simple, Double, Hyperbola: 36-55;19,27,63,78, Tri.,tuadri.: 73,140,163,164, 79,88,101,112,115,116,129,130, 174,203) 139,140,144,149,157,163,164, Fourier Development: 232 168,169,173,182,184,189,195, Fucia: 186 205,214,219 Functions with Discontinuous Hyperbolic Catenary: 182,184 Properties: 100-107 Hyperbolic Functions: 113-118 Hyperbolic Spiral: (see Spirals, Galileo: 12,65,66,81 Reciprocal) Gaussian Curve: 95 Hypocycloid: 81-85;1,63,71,87, Gears: 1,69,81,137,233 122,126,140,163,169,177,180, Gerono's Lamniscate: 203 182 Glissettes: 108.112;50,121,122, Hypotrochoids: (see Trochoids) 138,139,149,216 Goblet: 204 Ionic Volute: 213 Grandi: 237 Ingram: 127 Graphical solution of oubics: Instantaneous Center of Rota- 57,58 tion: 119-122;3,15,29,32,66, Great circle route! 230 73,85,153,158,176,209 Gregory: 238 Intrinsic Equations: 123-126; Growth, Law of: 94 92,180 Gudermann: 113 Inversion: 127-134; 63 Gudermannian: 115 Involutes: 135.137;13,20,66,68, Guillery: 69 85,87,125,126,155,156,164, 176,182,183,208,209,222 Halley: 207 Isolated point: 192,197,200, Harmonic Analysis: 230 202 Harmonic Motion: 6,67,229 Isoptic Curves: 138-140;69,85, Harmonic Section: 42 121,213 Hart: 131 2)42 INDEX

Jones: 210 Mechanisms, quick return: 183 Menaechm,1: 36 Kakeya: 72 Mercator: 118,250 Kampyle of Eudoxus: 174,204 Mersenne: 65,81 Kappa Curve: 174,204,205,222 Minimal Surfacer,: 13,183 Kelvin: 127,141,142 Monge: 56 Kieroid: 141.142;29,33,219 Montucla: 69 Kite: 158 Morley: 171 Motion, harmonic: 6,229 Lagrange: 15,67,75 Motion, line: 84,132,158,210, Lambert: 113 234 Lame Curve: 87,164,204 MUller: 225 Law of Growth (or Decay): 94 Multiple point: 20,192,197,199, Law of Sines: 225 200,202 Least area: 72 Leibnitz: 56,68,155,175,186, Napier: 93 221,238 Napkin ring: 17 Lemniscate, Bernoulli's: 143- Nasir: 234 147;9,10,63,130,157,163,168, Nautilus, septa of: 208 205,214 Neil: 186 Lemniscate, Gerono's: 203 Nephroid: 152.154;17,73,84,87, Light rays: 15,86 126 Limacon of Pascal: 148.151;5,7, Newton: 28,51,56,60,67,68,81, 16,31,108,110,121,130,139,140, 175 163,234,235 Nicomedes, Chonchoid of: 31-33; Line motion: 84,132,158,210,234 108,142 Linkages: 6,9,28,51,132,146,151, Node: 192,197,199,200 158,183 Normal Curve: 95,96 Liouville: 221 Nonnals: 91 LissajousCurves(see Bowditch Curves) Optics: 40,203 Lituus: 169,212,213 Orthocenter: 22 Logarithmic Spiral: (see Spirals, Orthogonal trajectory: 2',:3 Equiangular) Orthoptic: 3,73,138,139,149 Loria: 186 Orthotomic: 15,20,87,160 Lucas: 171 Osculating circle: 60,63 Osculinflexion: 198,199,200,202 Maclaurin: 143,160,163,182,205, Ovals: 131,149,203 213 Mapping: 118,223,230 Palm Stems: 204 Maxwell: 136,175 Paper Folding: 50,78, Mayer: 113 Pappus: 25 Mechanical Inversors: 131 Parabola: 36-551,12,13,19,20, Mechanical solution of cubic: 57 27,29,61,64,73,76,79,80,87,88, INDEX 243

91,111,112,129,136,138,139, Pursuit Curve: 170-171 140,149,156,157,161,163,164, Pyramid: 204 16+1,169,173,176,1, 2,183,14,7, 189,214,216,219 Quadratrix of Hippies: 204 Parabola, Cubic: (see Cubic ',uadrifolium: 140,163 Parabola) Quetelet: 15,127,160 Parabola, Semi - cubic: (see Semi. Quick return mechanism: 183 cubic Parabola) Parallel Curves: 155 - 159;79 Radial Curves: 172-174;69,73, Parallelogram, Crossed: 6,131, 222 158,183 Radical Axle: 22 Pascal: 36,65,148; Theorem of: Radical Center: 22 45 ff. Reciprocal Spiral: (see Spirals, Pearls of Sluze: 56,204 Reciprocal) Peaucellier cell: 10,28,52,131 Reflection: (see Caustics) Pedal Curves: 160-165:5,905,29, Refraction: 69, (see Caustics) 65,72,79,85,136,138,144,149, RhodcLeae: (see Roses) 167,179,182,203,207,209,214, Rhumb line: 118 219 Riccati: 113 Pedal Equations: 166-169;162, Ridolphi: 235 177,213 Roberval: 65,66,148 Pendulum, Cycloidal: 68 Roemer: 1,81,233 Pilaster: 204 Roses: 85,163,174,216,235 (also Pink: 186 see Trochoids) Piriform; 204 Roulettes: 175-185;13,29,65,73, Pivot, Schiele's: 222 79,110,135,136,207,212,222, Poinsot's Spiral: 204 233,235 (see Trochoide) Pointe, Singular: 192,199,200, Sacchi: 169 202 Sail, section of: 14 Polars: 41,42,43,44,133 Schiele's pivot: 222 Polynomial Curves: 64,89,194,198 Secant property: 21 Polynomial Curves, Semi-: 61,87, Semi-cubic Parabola: 186-187;61, 201 87,157,192,201 Power of a point: 21 Semi-polynomials: 61,87,201 Probability Curve: 95 Semi - trident: 204 Projection, Mercator's: 118; Septa of the Nautilus: '08 Orthogonal: 229; Series: 117,227,228,232,238 Orthographic: 211; Serpentine: 204 Stereographic: 207, Shoemaker's knife: 25 219 Sierpinski: 107 Prolate Cycloid: 65,69 (see Similitude: 22 Trochoids) Simeon line: 72 Pseudo - sphere: 223 the Curve: 225,229,230,231,232 Pseudo- witch: 238 Sines, Law of: 225 244 INDEX

Singular points: 62,192,197, Sturm: 26 199,200,202 Suardi: 235 Singular solutions: 75 Supercharger: 210 Sinusoidal Spirals: (see Swing and Chair: 204 Spirals, Sinusoidal) Syntractrix: 204 Sketching: 18e-205;155 Slope: 191 Tangent Construction: 3,13,29, Slot machirm: 96 32,41,44,46,66,75,85.119,139, Sluze, Pearls of: 204 145,150,155,168,214,222 Snowflake Curve: 106 Tangents at origin: 191,192 Soap films: 13,183 Tautochrone: 67,85 Spirals: 206-216 Taylor: '75 Spirals, Terquems 160 Archimedean: 20,136,164,169, Torus: 9,204 209,210,211,212 Tractory: (see Tractrix) Cornu's: (see Spirals, Euler) Tractrix: 221-224;13,63,87,126, Cotes'; 72,169,215,216 137,174,182,204,212 Equiangular: 20,63,87,126, Trains: 24 136,163,169,171,175,206, Trajectory, orthogonal: 223 207,208,209,211,216 Trammel of Archimedes: 3,77)108) Euler: 136,215 120,234 Fermat's: (see Spirals, Para- Transition curve: 56,215 bolic) Trident: 205 Hyperbolic: (see Spirals, Trifolium: (see Folium) Reciprocal) Trigonometric functions: 225-232 Parabolic: 169,212,213 Trieection: 33,36,58,205 Poinsot's: 204 Trisectrlx: 149,163,205,205 Reciprocal: )92,210,211, Trochoids: 253-236;120,122,138, 212,216,222 159,148,176,204 Sinusoidal: 20,63,139,140, Trophy: 204 144,161,162,163,168,205, Tschirnhausen: 15,152,203,205, 213,214 214 Spiral Tractrix: 137 Tucker: 172 Spiric Lines of Perseus: 204 Tulip: 186 Spring, elastic: 215 Tunnel: 204 Squaring the circle: 36,209 Twisted Bow: 204 Standard deviation: 96 Steiner: 24,127,179 Unduloid: 184 Step function: 232 Urn: 204 Stereographic projection: 207, 219 Varignon: 211 Strophoid: 217-220129,129,141, Versiera: (see Witch of Agnesi) 142,163,205 Versorio: (see Witch of Agnesi) Stubbs: 127 Vibration: 68,230,231,232 INDEX 245

Viviani's Curve: 205,219 Weierstraes: 113 Volute: 213 Weierstraos function: 107 Von Koch Curve: 106 Whewell: 87,123,124,125,126,180 Witch of Agnes': 237-238; 205 Wallis: 65 Wren, Sir Christopher: 66 Watt: 143 x y Wave theory: 230 yu x : 205